OF-  THK 

University  of  California. 


GIKT  OP^ 


Accession  H5970  ^^^^^ 


HEATH'S 


COMPLETE  PRACTICAL 


AEITHMETIC 


BY 

CHARLES   E.  WHITE 

SYRACUSE,  N.Y. 


BOSTON,  U.S.A. 

D.   C.   HEATH   &   CO.,   PUBLISHERS 

1901 


Copyright,  igoi, 
By  D.  C.  Heath  &  Co. 


6 


PREFACE. 

In  preparing  this  book,  the  aim  has  been  to  make  it 
complete,  so  that  it  might  cover  with  sufficient  fulness  the 
topics  usually  required  in  common  and  grammar  schools. 

The  aim  also  has  been  to  make  the  work  practical,  i.e.  to 
use  only  such  rules,  definitions,  solutions,  and  problems  as 
would  accustom  the  learner  to  reason,  compute,  and  esti- 
mate with  the  same  facility  and  economy  of  thought  with 
which  business  men  reason,  compute,  and  estimate  in  prac- 
tical life. 

Great  care  has  been  exercised  to  omit  the  merely  theo- 
retical and  the  unpractical,  as  well  as  problems  that  are  too 
hard  for  the  grades  for  which  the  book  is  intended. 

The  relations  of  the  inductive  work,  the  definition,  the 
solution,  the  rule,  are  such  that  the  learner,  with  a  mini- 
mum of  help,  should  easily  master  the  difficulties  through 
the  logical  progression  of  the  steps  from  the  known  to  the 
unknown. 

While  care  has  been  exercised  in  selecting  a  great  variety 
of  practical  business  problems  and  in  arranging  them  pro- 
gressively, the  development  of  mental  power  has  been  kept 
constantly  in  view. 

The  arrangement  of  this  book  is  topical,  but  subjects 
previously  studied  are  kept  fresh  in  the  minds  of  the  pupils 
by  frequent  carefully  prepared  reviews. 


IV  PREFACE. 

Questions  of  relation,  as  treated  in  connection  with  divi- 
sion of  fractions,  will  be  found  helpful  in  overcoming  a 
group  of  difficulties  which,  in  the  author's  experience,  are 
trying  to  all  children.  When  these  are  mastered  by  the 
learner,  he  will  later  have  little  or  no  difficulty  with  Per- 
centage and  its  applications. 

^  The  practice  of  referring  percentage  problems  back  to 
the  original  questions  of  relation  has  proved  highly  suc- 
cessful in  the  experience  of  many  teachers. 

A  wide  variety  as  well  as  a  great  number  of  topical 
review  and  miscellaneous  problems  are  given  in  the  last 
part  of  the  book. 

Thanks  are  due  to  the  various  superintendents  of  city 
schools  who  have  kindly  furnished  copies  of  recent  exami- 
nation questions,  which  largely  constitute  the  topical  review 
of  this  book. 

The  author  has  also  received  invaluable  aid  from  many 
leading  educators,  all  of  whom  he  desires  to  thank  most 
cordially. 

C.  E.  W. 

Syracuse,  N.Y., 
January  24, 1901. 


CONTENTS. 

TAQZ 

Notation  and  Numeration 1 

Arabic  Notation 2 

llomau  Notation 9 

Notation  of  United  States  Money  .        .         .        .         .        .10 

Addition 12 

Subtraction 22 

Multiplication 31 

Division 42 

Short  Division 48 

Long  Division 51 

Indicated  Operations 54 

Principles  of  Division 55 

Factors 61 

Cancellation 63 

Greatest  Common  Divisor 65 

Least  Common  Multiple 68 

Common  Fractions 73 

Ked action  of  Fractions 75 

Addition 83 

Subtraction 86 

Multiplication 89 

Division ....  93 

The  Three  Questions  of  Relation 98 

Review 101 

Decimal  Fractions 108 

To  read  and  write  a  Decimal 109 

Reduction  of  Decimals 110 

V 


si 


VI  CONTENTS. 

PAGE 

Addition  of  Decimals     .        . 112 

Subtraction  of  Decimals 113 

Multiplication  of  Decimals 114 

Division  of  Decimals 116 

To  divide  by  10,  100,  1000,  etc 117 

Parts  of  100  and  1000 118 

Aliquot  Parts  of  $1.00 119 

Review  of  Decimals 121 

Accounts  and  Bills   .         .        .        .         .        .         .         .        .  125 

Compound  Numbers 135 

Linear  Measure 135 

Surveyor's  Measure 136 

Square  Measure 136 

Cubic  Measure 136 

Liquid  Measure 137 

Apothecaries'  Fluid  Measure 137 

Dry  Measure .        .         •  137 

Avoirdupois  Weight 137 

Troy  Weight 138 

Apothecaries'  Weight 138 

Comparison  of  Weights *      .         .  138 

Measure  of  Time 138 

Circular  Measure 139 

Federal  Money 140 

English  Money .        .  140 

Counting  Table 141 

Paper  Table 141 

Reduction  Descending 141 

Reduction  Ascending 143 

Reduction  of  Denominate  Fractions  to  Integers  of  Lower 

Denominations 147 

eduction  of  Denominate  Numbers  to  Fractions  of  Higher 

Denominations 148 


/-^ 


CONTENTS.  Vii 

PAGE 

To  find  what  Part  One  Denominate  Number  is  of  Another  .  150 

Addition  of  Compound  Numbers 151 

Subtraction  of  Compound  Numbers 153 

Difference  between  Dates 155 

Multiplication  of  Compound  Numbers 156 

Division  of  Compound  Numbers 157 

Miscellaneous  Problems 161 

Measurements,  Surfaces 164 

Carpeting  Eooms 169 

Plastering  and  Painting 171 

Papering  Walls      .........  173 

Board  Measure 175 

Miscellaneous  Problems 176 

Measurements,  Volumes 178 

Wood  Measure 181 

Capacity  of  Bins  and  Cisterns 182 

Longitude  and  Time  .........  183 

Standard  or  Railroad  Time 186 

The  Metric  System 191 

Linear  Measure 192 

Surface  Measure 194 

Volume  Measure 19'/ 

Capacity  Measure 198 

Measures  of  Weight        .         . 200 

Review  Questions 201 

General  Review 202 

Percentage 210 

Profit  and  Loss 220 

Commission 223 

Insurance 227 

Trade  Discount 233 

Taxes 235 

Duties 237 


Vm  CONTENTS. 

'  PAGE 

Review  Questions 238 

Miscellaneous  Review  of  Percentage      '.....  239 

Simple  Interest 246 

The  Six  Per  Cent  Method 247 

Exact  Interest 251 

Problems  in  Interest .  252 

Promissory  Notes 255 

Partial  Payments  .         .         . 258 

Compound  Interest 262 

Review  of  Interest 263 

Discount • 267 

True  Discount 267 

Bank  Discount 269 

To  find  the  Face  of  a  Note  when  Proceeds,  Time,  and  Rate 

are  known 275 

Review  of  Discount 276 

Stocks  and  Bonds 278 

Bonds 281 

Average  of  Payments 286 

Ratio  and  Proportion 292 

Ratio .        .         .  292 

Proportion 294 

Simple  Proportion          .         .         .         .        .         .        .        .  295 

sQompound  Proportion 298 

j  ~T  (   Partnership 301 

/      Involution 307 

t--^VOLUTION 309 

Evolution  and  Involution         .         .         .         .        .         .        .  310 

Square  Root 310 

Right-angled  Triangles  .        .        .        ...        .        .  315 

Similar  Surfaces 317 

Cube  Root 318 

Similar  Solids .        .        .325 


-'I  Pi 


CONTENTS.  IX 

PAGB 

Questions        ....*. 326 

General  Review 327 

Topical  Review 340 

Common  Fractions 346 

Decimals 349 

Denominate  Numbers 363 

Percentage .        .        ,  360 

Interest  and  Discount 36§ 

Proportion  and  Partnership 376 

Involution  and  Evolution 379 

Miscellaneous  Problems  ........  381 

Mensuration        .         .         .         .         .        .         .         .         .         .  391 

Surfaces 392 

Solids 393 

Pyramids  and  Cones 394 

APPENDIX. 

Mariners'  Measures 397 

Surveyors'  Linear  Measure 397 

Surveyors'  Square  Measure     .......  398 

Government  Lands     .........  398 

Miscellaneous  Measures  of  Weight 399 

Apothecaries'  Fluid -Me  a  sure 401 

Business  Forms   ..........  402 

Computing  Taxes 404 

Table  of  Legal  Rates  of  Interest        .         .         .         .        .  405 

Exchange 406 

Domestic 406 

Foreign 409 


\:  \  B  R  A  rf^ 
UNX/EHSITY 


COMPLETE  PEACTICAL  AEITHMETIC. 


3>©<C 


NOTATION   AND   NUMERATION. 


1.  A  Unit  is  one,  or  one  thing;  as  one,  one  orange,  on© 
dollar. 

2.  A  Number  is  that  which  tells  how  many,  and  consists 
of  one  or  more  units. 

3.  The  Unit  of  a  Numoer  is  one  of  its  units.     The  unit 
of  seven  men  is  one  man.     The  unit  of  seven  is  one. 

4.  Numbers  having  the  same  unit  are  Like  Numbers. 
Thus,  3,  4,  and  5 ;  6  apples,  4  apples,  and  3  apples ;  are 

like  numbers. 

5.  A  number  not  applied  to  any  particular  object  is  an 
Abstract  Number ;  as  6,  11,  15. 

6.  A  number  that  is  applied  to  any  particular  object  is 
a  Concrete  Number ;  as  6  men,  11  pounds,  15  days. 

7.  An  Integer  is  a  whole  number. 

8.  The   expression   of  numbers   by  figures  or  letters  is 
called  Notation. 

9.  The  expression  of  numbers  by  figures  is  called  Arabic 
Notation.     It  was  used  by  the  Arabs. 

1 


^  NOTATION   AND   NUMERATION. 

10.  The  expression  of  numbers  by  letters  is  called  Roman 
Notation.     It  was  first  used  by  the  ancient  Romans. 

11.  The  reading  of  numbers  is  called  Numeration. 

12.  Figures  are  the  characters  used  to  express  numbers. 

ARABIC  NOTATION. 

In  the  Arabic  notation  ten  different  figures  are  used; 
they  are : 

0,  1,       2,         3,       4,       5,      6,      7,        8,        9. 

Naught,    One,    Two,   Three,    Four,   Five,   Six,   Seven,    Eight,    Nine. 

NoTi:. — Naught  is  sometimes  called  zero,  and  cipher.  The  other 
niue  figures  are  called  digits,  from  a  Latin  word  meaning  fingers. 

13.  Naught  used  alone  expresses  no  units,  or  nothing. 
The  other  nine  figures  express  the  number  of  units  indi- 
cated by  their  names. 

14.  To  express  a  number  larger  than  nine,  two  or  more 
figures  are  written  side  by  side. 

15.  A  figure  standing  alone  expresses  one  or  more  units. 

16.  When  figures  stand  side  by  side,  the  right-hand  figure 
expresses  units,  the  next  tens,  the  next  hundreds,  etc. 

17.  The  value  of  a  figure,  without  regard  to  its  place,  is 
its  Simple  Value.  The  value  of  a  figure  with  reference  to 
its  place  in  a  number  is  its  Local  Value. 

Note.  — In  the  number  6555,  the  simple  value  of  each  figure  is  5, 
The  local  value  of  the  right-hand  figure  is  5.  Of  the  second,  50.  Of 
the  third,  500.     Of  the  fourth,  5000. 

18.  Figures  in  the  units'  place  express  units  of  the  first 
order  ;  those  in  the  tens'  place  express  units  of  the  second 
order  ;  those  in  the  hundreds'  place,  units  of  the  third  order  ; 
etc. 


ARABIC   NOTATION.  6 

19.  The  units  of  the  second  order,  or  tens,  are  ten,  twenty, 
thirty,  forty,  fifty,  sixty,  seventy,  eighty,  ninety. 

One  ten  is  written  10 ;  two  tens,  20 ;  three  tens,  30 ;  etc. 

20.  The  numbers  between  10  and  20  are  one  ten  and  one 
unit,  or  11 ;  one  ten  and  two  units,  or  12 ;  one  ten  and  three 
units,  or  13 ;  one  ten  and  four  units,  or  14 ;  etc. 

21.  Two  tens  and  one  unit  is  21 ;  five  tens  and  six  units 
is  56 ;  nine  tens  and  four  units  is  94  ;  etc. 

Read  the  following : 

1.  16       4.  29  7.  33 

2.  84       5.  65  8.  87 

3.  98       6.  77  9.  49 

Write  in  figures : 

13.    Seventeen.         14.    Twenty-five. 

16.  Seven  tens  and  two  units. 

17.  Nine  tens  and  nine  units. 

18.  Three  tens  and  no  units. 

22.  Write  all  the  numbers  between  nine  and  twenty; 
between  twenty  and  forty. 

23.  Write  all  the  numbers  between  sixty  and  ninety. 

24.  Write  four  units  of  the  second  order  and  six  units  of 
the  first  order. 

25.  What  number  is  expressed  by  writing  seven  units  of 
the  second  order  and  five  units  of  the  first  order  ? 

22.  Numbers  having  three  figures  are  written  with  the 
hundreds  at  the  left  of  the  tens.  Figures  in  the  third  place 
are  called  units  of  the  third  order. 

Thus,  482  is  read  four  hundred  eighty-twa 


10.    59 

11.    72 

12.    99 

15. 

Thirty-four. 

19. 

Eighty-four. 

20. 

Fifty-two. 

21. 

Sixty-one. 

32.  729 

35.  Ill 

33.  984 

36.  504 

34.  796 

37.  600 

4  NOTATION  AND   NUMERATION. 

Read  tlie  following : 

26.  384  29.    m6 

27.  583  30.    840 

28.  654  31.    972 

38.  Seven  hundred  fifty-six. 

39.  Six  hundred  fifty-three. 

40.  Five  hundreds,  eight  tens,  six  units. 

41.  9  hundreds,  5  tens,  3  units. 

42.  7  hundreds,  0  tens,  0  units. 

43.  4  hundreds,  0  tens,  6  units. 

44.  Nine  hundred  ninety. 

45.  Nine  hundreds,  nine  tens,  9  units. 

46.  8  units  of  the  third  order,  6  units  of  the  second  order, 
2  units  of  the  first  order. 

47.  Five  units  of  the  3d  order,  6  units  of  the  1st  order. 

23.  Units  of  the  fourth  order  are  written  at  the  left  of 
hundreds'  place. 

Thus,  4876  is  read  four  thousand  eight  hundred  seventy- 
six,  or  4  thousands,  8  hundreds,  7  tens,  6  units. 

Eead  the  following : 

48.  2876  51.    9999  54.    3970 

49.  3972  52.    5063  55.   4006 

50.  8763  53.    6205  56.   5321 
Write  in  figures : 

57.  Two  thousand  nine  hundred  twenty-six. 

58.  5  thousands,  6  hundreds,  4  tens,  8  units. 

59.  6  thousands,  0  hundreds,  0  tens,  2  units. 

60.  Six  thousand  four  hundred  eight. 

61.  Six  thousand  eight. 

62.  Six  thousand  eighty. 

63.  5  units,  5  tens,  5  hundreds,  5  thousands. 


ARABIC   NOTATION.  O 

24.  On  the  same  principle  units  of  the  fifth  order  occupy 
the  fifth  place,  and  are  called  teyi-tliousands.  Units  of  the 
sixth  order  occupy  the  sixth  place,  and  are  called  hundred- 
thousands.  Units  of  the  seventh  order  occupy  the  seventh 
place,  and  are  called  millions. 

25.  Ten  units  of  any  order  make  one  unit  of  the  next  higher 
order. 

26.  The  first  ten  orders  of  units  are  shown  in  the  fol- 
lowing 

TABLE, 

lOtn    9ttL    8th.    TttL    Gtli    5tli    4tli    3d      2d     1st 


s  i  I"  S 

UJ  2  <  XJ 

5^  £  »  £  - 

9  ^  2  9  «> 

z  z  O  z  z 

3  U  I  3  UJ 


27.  For  convenience  in  reading  and  writing  numbers  they 
are  separated  into  groups  of  three  figures  each,  q^Wq^  periods. 
Each  group  takes  the  name  of  its  right  hand  order  of  units ; 
thus,  the  first  group  is  the  group  of  units,  the  second  of 
thousands,  the  third  of  millions,  the  fourth  of  billions. 

The  comma  is  used  to  separate  the  groups. 

Thus,  in  the  number  624,503,275,320, 

the   first    group  is  320  units, 
the  second  group  is  275  thousands, 
the   third   group  is  503  millions, 
the  fourth  group  is  624  billions. 

The  above  number  is  read  as  follows : 

624  billion,  503  million,  275  thousand,  320. 
Note.  —  In  reading  numbers  the  last  group-name  is  always  omitted 


t)  NOTATION    AND   NUMERATION. 

PRINCIPLES  OP  NOTATION. 

28.  1.  Ten  units  of  any  order  are  equal  to  one  unit  of  the 
next  higher  order. 

2.  The  value  of  a  figure  is  increased  teyifold  by  removing 
it  one  place  to  the  left,  and  decreased  tenfold  by  removing  it 
one  place  to  the  right. 

The  following  table  shows  the  grouping  of  the  orders  into 
periods : 


=!  2  5  z  z  < 

go  S  o  o  5 

oil  f2  5g  ii  ^^ 

-§^  2gi  25-  S5^  So 


't/vQL  "-1—  "-;^o  "-«a  u-.co  u- 

Q?Q  Q^H  Q"?^  qS2  Qh3  Qco£ 

zz<  zz=i  zz^  zzH  zzo  zzt 

3li]3  Zi    \ii    a.  DuJ=  DuJ—  3LiiZ  3lJZ 

Il-Cr  Ihh-  IhCQ  IHS  XHI-  XI-3 


32,        406,       398,       040,        324,       763 


Groups,        Gtli  5th.  AtYi  3d  2d  1st 

Name,  Quadrillions  Trillions     Billions     Millions    Thousands    Units 

The  above  number  is  read  32  quadrillion,  406  trillion, 
398  billion,  40  million,  324  thousand,  763. 

Note  1.  — The  names  of  groups  above  quadrillions  are  quintillions, 
sextillions,  septillions,  octillions,  nonillions,  decillions,  etc. 

Note  2.  —  Each  group  except  the  one  at  the  left  must  contain 
three  figures. 

TO    BEAD   NUMBERS. 

29.    Rule. — Begin  at  the  right,  and  separate  the  numbers 
into  groups  of  three  figures  each,  using  the  comma. 
Begin  at  the  left,  and  read  the  number  in  each  group,  giving 

to  it  the  name  of  that  group. 
No  name  is  given  to  the  number  in  the  last  group. 


ARABIC   NOTATION. 


30.    Copy,  separate  into  groups,  and  read : 


1.  3896 

2.  26432 

3.  897063 

4.  20396 

5.  390403 

6.  704503 

7.  2987652 

8.  356293603 

9.  290030052 
10.  387523729842 


14.  500004 

15.  329000101 

16.  3424300000 

17.  800003000 

18.  29856323155824 

19.  1487603035006201 

20.  35601600 

21.  18008 

22.  180506 

23.  1658838 


11.  5030473694026  24.  200800 

12.  500320       25.  32004060 

13.  3400093 


26.  13087 

27.  716042 

28.  2730010 

29.  126003184 

30.  47250627  . 

31.  1002970 

32.  17042 

33.  14390023 

34.  11935079 

35.  4000030 

36.  29307070 

37.  14280643 


31.   Write  in  figures : 

1.  Thirty-six  million,  twenty-four  thousand,  two  hundred 
seventy-two. 

36,024,272. 

Solution,  —  Write  35  for  the  millions'  group,  following  with  a 
comma ;  then  write  the  24  in  the  thousands'  group,  prefixing  naught 
to  make  the  group  complete,  and  follow  it  with  a  comma.  This  is 
followed  by  272  in  the  units'  group. 


TO  WRITE  NUMBERS. 


32.  Rule.  —  Beginning  at  the  left,  write  the  figures  of  each 
group  in  their  proper  order,  filling  vacant  places  with 
ciphers.  Place  a  comma  after  each  group  before  writing 
the  following  group. 


8  NOTATION   AND   NUMERATION. 

Write  in  figures  : 

1.  Twenty-seven  thousand,  three  hundred  sixteen. 

2.  Eighty-four  thousand,  seven  hundred  twenty-six. 

3.  One    hundred    twenty-two    thousand,    one   hundred 
forty-five. 

4.  Two  hundred  thousand,  sixteen. 

5.  Eleven  thousand,  two. 

6.  Four  million,  six  hundred  eight  thousand,  three  hun- 
dred seventy-five. 

7.  Twenty-five  thousand,  three  hundred  eighty-seven 

8.  Nineteen  thousand,  seventeen. 

9.  Twenty-seven  million,  six  hundred  fifty-two. 

10.  Eighty   million,    six    hundred   nine   thousand,   four 
hundred  twenty-eight. 

11.  Four  hundred  thirty-six  thousand,  forty-one. 

12.  Six   hundred    twenty   million,   seventeen  thousand, 
four  hundred  seventy-seven. 

13.  One  hundred  fifty-seven  million,  six  hundred  eight 
thousand,  four  hundred  seventy-seven. 

14.  Write  a  number  containing  four  groups. 

15.  Six   hundred   four  million,  seventy-eight  thousand, 
nine  hundred  two. 

16.  Three  hundred  twenty-four  billion,  two  thousand,  six 
hundred  forty. 

17.  Six  hundred  thousand,  fifty-five. 

18.  Four  million,  three  hundred  six  thousand,  one  hun- 
dred eight.     Forty  thousand,  ten. 

19.  75  million,  136  thousand,  265. 

20.  356  billion,  208  million,  708  thousand,  16. 

21.  Five  billion,  five  million,  five  thousand,  five.     Four 
thousand,  four. 


ROMAN   NOTATION.  9 

22.  306  million,  20  thousand,  12.  Seventeen  million,  2 
thousand,  406. 

23.  Ninety-four  trillion,  sixteen  billion,  four  hundred  six 
million,  fifteen  thousand,  seven  hundred. 

24.  20  million,  20  thousand,  20.  Four  hundred  twenty- 
five  million,  seven  hundred  two  thousand,  one  hundred 
eighty-one. 

25.  60  billion,  40  thousand.     1  billion,  1  thousand,  1. 

ROMAN  NOTATION. 

33.  The  Eoman  system  of  notation  uses  seven  capital 
letters  to  express  numbers,  viz. : 

I,  V,  X,    L,     C,      D,        M, 
Values,  1,  5,  10,  50,  100,  500,  1000. 

All  other  numbers  are  formed  by  repeating  or  combining 
these  letters. 

34.  The  following  principles  are  used  in  expressing 
!koman  numbers : 

Principles.  —  1.   Eepeating  a  letter  repeats  its  value. 
Thus,  I  stands  for  one  ;  II  for  two ;  III  for  three ;  X  for 
ten ;  XX  for  twenty ;  XXX  for  thirty,  etc. 

Note.  —  Only  I,  X,  C,  and  M,  are  thus  repeated. 

2.  When  a  letter  is  placed  before  another  of  greater  value, 
its  value  is  taken  from  that  of  the  greater. 

Thus,  IV  stands  for  four ;  IX  for  nine ;  XIX  for  nineteen ; 
XL  for  forty ;  XC  for  ninety. 

3.  When  a  letter  is  placed  after  another  of  greater  value, 
their  values  are  united. 

Thus  VI  stands  for  six ;  XII  for  twelve ;  XV  for  fifteen ; 
XXXV  for  thirty-five ;  LV  for  fifty-five. 


10 


NOTATION   AND   NUMERATION. 


value  a  thousand-fold. 
L  for  fifty  thousand ; 


4.   A  dash  over  a  letter  increases  its 

Thus,  V  stands  for  five  thousand ; 
M  for  one  million. 

The  following  table  illustrates  the  use  of  Roman  letters 
in  forming  numbers : 


I . 

.   1 

XI  . 

.  11 

XXIV  . 

.  24 

C  . 

100 

II . 

.     2 

XII  . 

.  12 

XXIX  . 

.  29 

cc  . 

200 

Ill . 

.     3 

XIII  . 

.  13 

XXX  . 

.  30 

cccc  . 

400 

IV  . 

.     4 

XIV  . 

.  14 

XL  . 

.  40 

CD  . 

400 

V  . 

.     5 

XV  . 

.  15 

L  . 

.  50 

D  . 

500 

VI  . 

.     6 

XVI  . 

.  16 

LX  . 

.  60 

DCC  . 

700 

V^II  . 

.     7 

XIX  . 

.  19 

LXX  . 

.  70 

DC  . 

600 

IX  . 

.     9 

XX  . 

.  20 

LXXX  . 

.  80 

M  . 

1000 

X  . 

.  10 

XXI  . 

.  21 

xc  . 

.  90 

MD  . 

.  1500 

35.  Read  the  following : 

XXIX;  XXXV;  LXX;  XXXIX;  XLIV;  CXV ;  XCV; 
LXXXIX;  CXIV;  XCIV ;  XLIX;  CCCIX;  CDXIV; 
XDLXII;  MDCCCC. 

36.  Write  in  Roman : 

39,  63,  98,  161,  515,  654,  1560,  1899,  1902,  1040,  3762. 


UNITED   STATES  MONEY. 

37.    The  Dollar  Sign  is  $,  and  is  placed  before  the  number. 
Thus,  $26  is  read  26  dollars. 

•  Dollars  are  written  at  the  left  of  a  period  (.)  called  the 
Decimal  Point.  Cents  are  written  at  the  right  of  the  deci- 
mal point,  and  always  occupy  two  places. 

Mills,  or  tenths  of  a  cent,  occupy  the  third  place  at  the 
right  of  the  decimal  point. 

Twenty-four  dollars  sixty-five  cents  three  mills  is  written 
$24,653. 


UNITED    STATES   MONEY.  11 

Any  number  of  cents,  less  than  10,  requires  a  naught  be- 
tween it  and  the  decimal  point. 

Thus,  one  dollar  and  8  cents  is  written  ^1.08. 

Copy  and  read : 

1.  $25.34  6.  .flj.03  11.  $204,865 

2.  $74.98  7.  $2,843  12.  $384,467 

3.  $25.09  8.  $200,504  13.  $293,062 

4.  $87.15  9.  $192,003  14.  $398,405 

5.  $58.16  10.  $715.38  15.  $294,066 

Write  the  following : 

1.  Sixteen  dollars,  fifteen  cents.  Nineteen  dollars, 
seventy-five   cents.     Sixty-one  dollars,  twenty-eight  cents. 

2.  Twenty-five  dollars,  twenty  cents.  Two  hundred  dol- 
lars, eight  cents.     24  dollars,  5  cents. 

3.  1864  dollars,  11  cents.  87  dollars,  9  cents.  28  dol- 
lars, 28  cents.     19  dollars,  1  cent. 

4.  8  dollars,  5  cents,  5  mills.  12  dollars,  16  cents,  6 
mills.  4  dollars,  4  cents,  4  mills.  30  dollars,  30  cents, 
3  mills. 

5.  306  dollars,  10  cents,  4  mills.  806  dollars,  20  cents, 
2  mills.     1349  dollars,  9  cents,  9  mills. 

6.  85,600  dollars,  20  cents,  1  mill.  28  dollars,  7  cents, 
7  mills.     28  dollars,  7  mills. 

7.  1840  dollars,  4  mills.  268  dollars,  15  cents,  9  mills. 
84  dollars,  84  cents,  8  mills. 


ADDITION. 


38.  1.   How  many  apples  are  4  apples  and  2  apples  ? 

2.  How  many  pencils  are  3  pencils  and  4  pencils  ? 

3.  Willie  had  5  cents  and  his  uncle  gave  him  4  cents. 
How  many  had  he  then  ? 

4.  5  books  and  3  books  are  how  many  books  ? 

5.  How  many  oranges  are  5  oranges  and  2  oranges  ? 

6.  How  many  are  6  and  3  ?     7  and  4  ?     3,  2,  and  4  ? 

7.  How  many  books  are  2  books,  4  books,  and  5  books  ? 

39.  Addition  is  the  process  of  uniting  two  or  more  num- 
bers into  one  sum. 

40.  The  result  obtained  by  adding  is  called  the  Sum  or 
Amount. 

41.  The  Sign  of  Addition  is  an  upright  cross,  +.  It  is 
called  plus  and  is  sometimes  placed  between  numbers  to  be 
added. 

Thus,  5  +  4  is  read  5  plus  4,  and  means  that  5  and  4  are 
to  be  added. 

42.  The  Sign  of  Equality  is  two  short  horizontal  lines,  and 
means  equals,  or  equal  to. 

Thus,  5  +  3  =  8  is  read  5  plus  3  equals  8,  or  5  and  3 
are  8. 

12 


DRILL   IK   ADDITION.  _v  13 


DRILL  IN  ADDITION. 


43.    The  following  are  all  the  combinations  of  two  num- 
bers from  1  to  9. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

2 

2 

2 

2 

2 

2 

2 

3 

4 

5 

6 

7 

8 

9 

3 

3 

3 

3 

3 

3 

3 

4 

5 

6 

7 

8 

9 

4 

4 

4 

4 

4 

4 

5 

6 

7 

8 

9 

5 

5 

5 

5 

5 

6 

7 

8 

9 

6 

6 

6 

6 

7 

8 

9 

7 

7 

7 

8 

9 

8 

8 

9 

9 

Oral. 

44.  1.  John  found  4  eggs  in  one  nest  and  6  in  another. 
H^w  many  eggs  did  he  find  ? 

2.  James  had  7  cents  and  found  4  more.  How  many- 
cents  did  he  then  have  ? 


14  ADDITION. 

3.  If  Mary  paid  10  cents  for  a  tablet,  5  cents  for  a  pen- 
cil, and  3  cents  for  pens,  how  much  did  she  pay  for  all  ? 
5  +  3  =  ? 

4.  In  a  pasture  there  are  6  black  horses,  6  bay  horses, 
and  3  white  ones.  How  many  horses  are  in  the  field? 
6+5+3=? 

5.  There  are  9  yellow  apples  in  one  basket  and  5  red 
ones  in  another  basket.     How  many  apples  in  both  baskets  ? 

6.  I  spent  4  dollars  for  a  chair,  6  dollars  for  a  table,  and 
3  dollars  for  a  lamp.     How  much  money  did  I  spend  ? 

7.  Edward  had  9  cents  in  the  bank  and  put  in  9  cents 
more.     How  many  cents  did  he  then  have  in  the  bank  ? 

8.  3  +  4  +  6  =  ?    4  +  5  +  3  =  ? 

9.  John  caught  5  trout  on  Monday,  6  on  Tuesday,  and  4 
on  Wednesday.     How  many  trout  did  he  catch  ? 

45.  Add  by  twos  from  0  to  28. 
Thus,  2,  4,  6,  8,  10,  12,  14,  etc. 

1.  Add  by  2's  from  1  to  31. 

2.  Add  by  3's  from  0  to  39.     From  1  to  40. 

3.  Add  by  4's  from  2  to  34.     From  0  to  30. 

4.  Add  by  4's  from  1  to  37.     From  3  to  38. 

5.  Add  by  5's  from  0  to  40.     From  1  to  36. 

6.  Add  by  5's  from  3  to  38.     From  4  to  39. 

7.  Add  by  6's  from  0  to  30.  From  1  to  49. 
8.-  Add  by  6's  from  4  to  64.  From  5  to  53. 
9.  Add  by  7's  from  0  to  70.     From  1  to  71. 

10.  Add  by  7's  from  2  to  72.     From  5  to  89. 

11.  Add  by  8's  from  0  to  80.     From  5  to  93. 

12.  Add  by  9's  from  0  to  90.     From  2  to  92. 


ORAL   EXERCISES.  15 


46.   Add. 

1.   4      2. 

2 

3. 

7 

4. 

7 

5.   6 

6. 

5 

7. 

6 

8.   4 

8 

6 

1 

4 

9 

9 

7 

9 

7 

5 

3 

5 

4 

8 

3 

6 

3 

4 

4 

8 

3 

6 

9 

5 

2 

3 

9 

3 

5 

9 

4 

3 

51  6  2  4  4  5  8 

9.   7  +  5  +  8  +  3  +  9  +  8+4  =  ? 

10.  9  +  6  +  8  +  4  +  7  +  3  +  5  +  3  =  ? 

Oral. 

11.  What  is  the  sum  of  43  and  24. 

12.  A  railway  train  ran  35  miles  the  first  hour,  and  40 
miles  the  second  hour.  How  many  miles  did  it  run  in  the 
two  hours  ? 

13.  In  a  certain  class  there  are  26  boys  and  35  girls. 
How  many  pupils  are  in  the  class  ? 

14.  A  newsboy  made  31  cents  on  Monday,  15  cents  on 
Tuesday,  and  52  cents  on  Wednesd^ay.  How  many  cents 
did  he  make  in  the  three  days  ? 

15.  Lucy  bought  a  pineapple  for  21  cents,  and  two  pounds 
of  sugar  for  11  cents.     How  much  did  she  pay  for  both  ? 

16.  A  farmer  had  three  flocks  of  sheep.  The  first  flock 
contained  26  sheep,  the  second  37,  and  the  third  41.  How 
many  sheep  had  he  ? 

17.  A  father  gave  25  cents  to  his  son,  and  20  cents  to  each 
of  his  two  daughters.  How  much  money  did  he  give  to  his 
three  children? 

18.  A  farmer  sold  four  jars  of  butter.  The  first  con- 
tained 24  pounds,  the  second  26  pounds,  and  the  third  and 
fourth  25  pounds  each.  How  many  pounds  of  butter  did 
he  sell  ? 


16  ADDITION. 

19.  In  a  certain  grove  there  are  45  maple  trees,  34  oak 
trees,  and  28  beech,  trees.     How  many  trees  in  the  grove  ? 

20.  There  are  45  cattle  in  each  of  three  pastures.  How 
many  cattle  in  the  three  pastures  ? 

Written. 

21.  What  is  the  sum  of  635,  726,  and  893  ? 

Solution.  —  Write  the  numbers  so  that  units  of  the  same  order 
shall  stand  in  the  same  column. 

The  sum  of  the  units'  column  is  34-6+5=14.    14  units 

are  equal  to  1  ten  and  4  units.     Place  the  4  units  under 

' -^6        the  units'  column,  and  add  the  1  ten  to  the  column  of 

893        tens. 

noK4  1  +  9  +  2  +  3  =  15,  the  sum  of  the  tens.     15  tens  are 

equal  to  5  tens  and  1  hundred.     Place  the  5  tens  under 

the  tens'  column,  and  add  the  1  hundred  to  the  hundreds'  column, 

1+8  +  7+6  =  22,  the  sum  of  the  hundreds.  22  hundreds  are 
equal  to  2  thousands  and  2  hundreds,  which  are  placed  under  the 
thousands'  and  hundreds'  columns.     Hence  the  sum  is  2254, 

Note  1.  —  The  columns  should  be  added  a  second  time,  beginning 
with  the  top,  and,  if  the  sums  are  not  alike,  this  should  be  repeated 
till  the  sums  agree. 

Note  2.  —  When  the  sum  of  any  column  is  10,  20,  or  any  number 
ending  with  naught,  the  naught  is  placed  under  the  column  added, 
and  the  other  number  added  to  the  next  column. 

Note  3.  — In  adding  do  not  name  the  numbers  after  the  first  one. 
Say  3,  9,  14,  instead  of  3  and  6  are  9  and  5  are  14. 

Note  4, —  In  adding  United  States  money,  write  the  numbers  in 
columns,  with  the  decimal  points  standing  in  a  vertical  line,  and  add 
as  above.  Place  the  decimal  point  in  the  sum  directly  under  the  points 
above,  and  prefix  the  dollar  sign. 

Rule.  —  Write  the  numbers  so  that  units  of  the  same  order 
shall  be  in  the  same  column. 
Beginning  at  the  right,  add  the  columns,  placing  the  units  of 
the  sum  under  the  column  added,  and  add  the  tens,  if  any, 
to  the  next  column. 

Write  the  entire  sum  of  the  last  column. 


WRITTEN   EXERCISES. 


17 


47.    Add  and  prove  : 


1.  2673 

2.  837 

3 

.  628 

4.  8063 

846 

2964 

4307 

259 

1025 

418 

526 

8264 

92 

3825 

8279 

1287 

837 

842 

428 

428 

642 

29 

4273 

3064 

5.  $26.43 

6.  $715.30   7. 

$  165.00  8, 

.  $  20.863 

18.75 

21.86 

8.429 

129.40 

2.93 

9.246 

113.82 

5208.00 

4.10 

10.163 

6.804 

.926 

128.06 

7.208 

39.625 

128.753 

663.13 

516.00 

11.31 

37.15 

28.00 

8,096 

476.203 

192.097 

9.   4286 

10. 

3184 

11. 

2306 

3804 

2929 

1242 

9273 

3641 

8936 

6518 

8207 

6084 

8274 

9243 

8346 

2936 

6041 

2920 

8142 

2938 

1289 

9370 

8465 

9016 

6425 

3721 

2914 

3184 

4936 

8563 

6293 

8749 

8472 

12.  178469 

13. 

729476 

14. 

428756 

738527 

835694 

937524 

592946 

209731 

129305 

846953 

569420 

747925 

362431 

182134 

165213 

234234 

234123 

243121 

18 


ADDITION. 


16.  $15,684 
17.326 
28.11 
39.487 
24.16 
29.312 
43.291 


16. 


$11,164 
21.178 
31.486 
23.12 
34.196 
29.394 
84.684 


17. 


16.98 
13.043 

1.29 
31.751 
48.006 
28.775 

8.44 


18 
19. 
20. 
21. 
22. 


$19,723,    $5.80, 


Add  2468,  9416,  7843,  6974,  1306. 
Add  395,  25682,  50600,  39,  48732. 
Add  639,  746,  892,  948,  769,  498. 
Add  2493,  79621,  98725,  16053,  972341,  28739. 
Add  $21.98,  $17,543,  $2.17,  $1554,  $.155,  $3.82, 
$  1.756. 

23.  Add  $39,412,   $17,694,   $34,006, 
$6.94,  $2.97,  $4.62. 

Add  and  prove : 

24.  2786  25. 
1947 
1838 
5287 
6496 
8035 
2956 
8375 
1698 
3263 

28.  The  sum  of  369,  298,  and  492,  added  to  the  sum  of 
1628,  38,  and  297,  equals  what  ? 

29.  Add  seventeen  thousand,  nine  hundred  six;  four 
thousand,  two  hundred  eighty-nine;  eight  hundred  twelve 
thousand,  seven  hundred  eight;  six  hundred  two;  forty- 
two  thousand,  nine  hundred  two ;  twelve  thousand. 


5278 

26.  43562 

27.  22879 

4492 

84601 

43012 

8913 

92873 

92874 

6472 

26461 

69154 

7258 

30725 

82738 

5603 

92837 

56425 

8450 

68154 

78997 

7921 

15210 

37684 

2864 

2835 

1872 

1342 

5832 

5832 

WRITTEN   EXERCISES.  19 

30.  Find  the  sum  of  eleven  thousand,  six  hundred  seven- 
teen; sixty-eight;  four  thousand,  twenty-five;  two  thou- 
sand, three  hundred  nine;  eighty-five  thousand. 

31.  A  merchant's  sales  were  $2963.84  in  January, 
$  1463.27  in  February,  $  3846.25  in  March,  and  $  2016.92 
in  April.  What  did  his  sales  amount  to  in  the  four 
months  ? 

32.  Find  the  sum  of  all  the  numbers  between  and 
including  167  and  174. 

33.  A  man  had  $  170  in  his  pocket,  which  was  $  70.75 
less  than  he  had  in  his  safe.  How  much  money  had  he  in 
the  safe? 

34.  Add  nine  dollars  six  cents ;  fifteen  dollars  seventy- 
two  cents;  sixty  dollars  eighty-seven  cents;  fifty-nine 
cents ;  four  dollars  five  cents ;  two  hundred  dollars  thirteen 
cents. 

35.  Find  the  cost  of  the  following  articles :  coal, 
$10.50;  sugar,  $4.38;  flour,  $13.72;  wood,  $5.28;  pork, 
$12.93;  beef,  $16.05;  potatoes,  $15.97;  apples,  $9,875; 
clothing,  $46,195. 

36.  The  expenses  for  one  year  for  a  family  of  four 
persons  were  as  follows :  table,  $  375 ;  fuel  and  light, 
$  125 ;  physician,  $48 ;  car  fare,  $25 ;  clothing  for  man,  $  95 ; 
for  wife,  $  125 ;  for  two  children,  $  75 ;  church  expenses, 
$  57 ;  newspapers  and  magazines,  $  28 ;  servant,  $  175 ;  all 
other  expenses,  $  148.  What  were  the  entire  expenses  for 
the  year  ? 

37.  A  man  bought  a  lot  for  $  2125,  erected  a  house  upon 
it  at  a  cost  of  $  5486,  paid  taxes  amounting  to  $  58,  and 
insurance  amounting  to  $  75.  He  .desires  to  sell  his  prop- 
erty at  an  advance  of  $  1200.  What  shall  he  ask  for  the 
house  and  lot  ? 


20  ADDITION. 

38.  Find  the  total  population  of  the  five  largest  cities 
in  the  world. 

39.  A  speculator  bought  potatoes  for  $  2680,  corn  for 
$  5870,  apples  for  $  1596,  wheat  for  $  7942,  oats  for  ^  6793, 
barley  for  $  3978,  and  sold  his  entire  purchase  at  a  profit 
of  $  2984.     What  did  he  receive  ? 

40.  A,  B,  C,  and  D  are  partners  in  the  dry  goods  business. 
A  has  put  in  $  5825 ;  B,  $  3246  more  than  A ;  C,  as  much  as 
A  and  B  together ;  and  D,  as  much  as  all  the  others.  What 
is  the  entire  capital  of  the  firm  ? 

41.  How  far  will  a  man  ride  a  bicycle  in  a  week  if  he 
travels  75  miles  on  Monday,  84  miles  on  Tuesday,  86  miles 
on  Wednesday,  100  miles  on  Thursday,  95  miles  on  Friday, 
and  101  miles  on  Saturday  ? 

42.  The  landing  of  the  Pilgrims  occurred  128  years  after 
the  discovery  of  America  by  Columbus ;  the  Declaration  of 
Independence  followed  in  156  years ;  Washington  was  made 
President  13  years  later  and  72  years  before  the  Civil  War 
broke  out.     In  what  year  did  the  Civil  War  begin  ? 

43.  The  population  of  Minnesota  in  1890  was  1,301,826, 
of  Iowa  610,070  more  than  Minnesota,  and  of  Missouri 
767,288  more  than  Iowa.  What  was  the  total  population  of 
the  three  states  ? 

44.  Por  how  much  must  I  sell  a  horse,  a  cow,  and  a  pig, 
that  cost  me  $  125,  $  40,  and  $  4  respectively,  if  I  gain  $  35 
on  the  horse,  $  16  on  the  cow,  and  $  1.50  on  the  pig  ? 

45.  Five  barrels  of  sugar  weighed  respectively  348,  327, 
354,  335,  and  329  pounds.  What  was  the  weight  of  the 
whole  ? 

46.  In  an  election,  A  received  27,423  votes,  B,  19,804, 
and  C,  5366  votes  more  than  both  A  and  B.  How  many 
votes  did  all  three  together  receive  ? 


WRITTEN   EXERCISES. 


21 


47.  I  have  $  12,450  invested  in  bonds,  $  15,850  in  busi- 
ness, $32,745  in  real  estate,  and  $4395  in  bank.  How 
much  do  I  have  in  all  ? 

48.  The  area  of  the  United  States  is  3,556,300  square 
miles,  of  Canada  3,470,000  square  miles,  and  of  Mexico 
767,000  square  miles.     What  is  their  total  area  ? 

49.  If  it  takes  1254  feet  of  lumber  to  build  a  sidewalk, 
2248  feet  to  repair  a  htmse,  25,235  feet  for  a  barn,  and  5160 
feet  for  a  fence,  how  many  feet  will  it  take  for  all  ? 

50.  Six  loads  of  hay  weighed  respectively  1942,  2126, 
2049,  1807,  1645,  and  2214  pounds.  How  many  pounds  of 
hay  in  all  ? 


51.  816    52.  7016 

53.  $16.75 

54.  $24,076 

391 

5609 

29.48 

18.92 

175 

2854 

45.22 

45.805 

762 

6327 

19.80 

7.19 

549 

4005 

32.15 

19.843 

433 

1963 

14.05 

3.457 

257 

842 

11.63 

80.76 

624 

396 

20.49 

22.40 

988 

1901 

10.72 

64.068 

8450 

8.14 

30.12 

55.  18324 

56. 

88632 

57.  $  29.845 

16309 

79045 

107.16 

92745 

129836 

95.423 

47528 

95207 

16.95 

9216 

47658 

75.09 

16127 

21030 

40.206 

4556 

19019 

148.75 

65901 

410803 

862.60 

18104 

36521 

104.494 

90027 

17084 

53.51 

51042 

SUBTRACTION. 


48.  1.  A  farmer  had  9  sheep  and  sold  4  of  them.  How 
many  sheep  had  he  left  ? 

2.  Annie  is  now  8  years  old.  How  old  was  she  5  years 
ago? 

3.  John  had  11  marbles  and  gave  away  6  of  them.  How 
many  had  he  left  ? 

4.  Henry  lives  10  miles  north  of  the  city,  and  James  7 
miles  north.     How  far  is  it  from  James's  house  to  Henry's? 

5.  There  are  12  pnpils  in  a  class.  Five  of  them  are 
boys.     How  many  are  girls  ? 

6.  A  boy  picked  9  quarts  of  cherries,  and  sold  6  quarts. 
How  many  quarts  had  he  left  ? 

49.  Subtraction  is  the  process  of  finding  the  difference 
between  two  like  numbers. 

50.  The  Minuend  is  the  number  from  which  we  subtract. 

51.  The  Subtrahend  is  the  number  subtracted. 

52.  The  result  in  subtraction  is  called  the  Difference  or 
Remainder. 

53.  The  Sign  of  Subtraction  is  a  short  horizontal  line  — . 
It  is  called  rainus,  and  when  placed  between  two  numbers 
signifies  that  the  second  is  to  be  subtracted  from  the  first. 
Minus  means  less. 

12  —  8  =  4  is  read  twelve  minus  (or  less)  8  are  4. 

22 


ORAL   EXERCISES.  23 

Oral. 

7.  Tom   is   20   years   old,  and   Jack   12.      How  much 
younger  than  Tom  is  Jack  ? 

8.  There  were  14  eggs  in  a  nest,  but  6  have  been  taken 
away.     How  many  eggs  are  left  ? 

9.  I  bought  a  picture  for  $  20,  and  sold  it  at  a  loss  of  $  5. 
What  did  I  sell  it  for  ? 

10.  What  number  taken  from  16  leaves  9  ? 

11.  A  girl  bought  a  bill  of  goods  amounting  to  f  11. 
She  gave  the  merchant  a  ten-dollar  bill  and  a  five-dollar 
bill.     How  much  change  should  she  receive  ? 

10-3 

13  -  5 

9-4 

14-7 

12-5 

16-7 

15-12 

17-7 

20-12 

18-12 

22.  10-f?  =  15     15-10  =  ?     9  +  ? 

23.  11  +  ?  =  1T     17-11  =  ?      8  +  ?  =  lT     17-8  =  ? 

24.  12  +  ?  =  18     18-12  =  ?     7  +  ?  =  20     20-7  =  ? 

25.  6  +  ?  =  15     15-6=?     5-f?  =  17     17-5  =  ? 

26.  Subtract  by  2's  from  40  back  to  0. 

27.  Subtract  by  3's  from  27  back  to  9. 

28.  Subtract  by  4's  from  30  back  to  10. 


12. 

12-8 

15-5 

13. 

15  - 10 

8-2 

14. 

7-2 

8-3 

15. 

16-9 

15-8 

16. 

10-3 

11-4 

17. 

18-5 

17-6 

18. 

17-10 

16-11 

19. 

19-5 

18-6 

20. 

20-10 

20-11 

21. 

18-10 

18-11 

16-10 

9-4 

12-6 

16-6 

10-6 

11-5 

13-6 

12-7 

13-6 

14-8 

15-8 

14-6 

14-- 2 

13-3 

16-8 

15-9 

20-13 

20-14 

18-13 

18-14 

?  =  14 

14-9  =  ? 

24  SUBTRACTION. 

29.  Subtract  by  o's  from  68  back  to  18. 

30.  Subtract  by  6's  from  90  back  to  30. 

31.  Subtract  by  7's  from  35  back  to  0. 

32.  Subtract  by  8's  from  49  back  to  1. 

33.  Subtract  by  9's  from  84  back  to  12. 

54.  Principles.  —  1.  Only  like  numbers  can  be  sub- 
tracted. 2.  The  sum  of  the  subtrahend  and  remainder 
must  equal  the  minuend. 

34.  Find  the  difference  between  987  and  563. 

Minuend,       987  Solution.  —  We  write  the  less  number  under 

Subtrahend  563  ^^^  greater,  with  units  of  the  same  order  in  the 

15         .     -I        7<r/  same  vertical  line. 

Remainder,  424  ^      .    ^        ^      ..    i          .       ..       i..  , 

^  3  units  from  7  units  leaves  4  units,  which  we 

place  under  the  units'  column ;  6  tens  from  8  tens  leaves  2  tens,  which 
we  place  under  the  tens'  column  ;  5  hundreds  from  9  hundreds  leaves 
4  hundreds,  which  we  place  under  the  hundreds'  column. 

Proof. —  If  the  sum  of  the  subtrahend  and  the  remainder 
equals  the  minuend,  the  work  is  right. 

Thus,  563  +  424  =  987.     Hence  the  result  is  correct. 

Subtract  and  prove : 

35.  896  39.      687  43.    7698  47.   $8,567 
474                   234                 5432  5.453 


36. 

624 

40. 

9844 

44. 

8649 

48.  $.689 

513 

7631 

5437 

.365 

37. 

988 

41. 

4687 

45. 

$  85.56 

49.  f.894 

372 

789 

42. 

3471 
8894 

46. 

43.43 

.072 

38. 

$  75.45 

674 

6523 

61.34 

ORAL   EXERCISES.  25 

50.  A  drover  bought  878  sheep  and  sold  453  of  them. 
How  many  had  he  left? 

51.  I  bought  a  farm  for  $5976  and  sold  it  for  $6453. 
How  much  was  my  profit  ? 

52.  John  was  born  in  1884,  and  Jennie  in  1896.  How 
much  older  is  John  than  Jennie  ? 

55.    Oral. 

53.  Bought  an  overcoat  for  $25  and  shoes  for  $6.  How 
much  more  than  the  shoes  did  the  coat  cost  ? 

54.  A  lady  gave  a  twenty-dollar  bill  in  payment  for  a 
fifteen-dollar  purchase.  How  much  change  should  she  re- 
ceive ? 

55.  A  miller  bought  oats  for  38  cents  a  bushel  and  sold  at 
42  cents  a  bushel.     What  was  his  profit  per  bushel  ? 

56.  A  newsboy  paid  25  cents  for  his  papers  and  sold  them 
for  50  cents.     What  was  his  profit  ? 

57.  Henry  deposited  in  the  bank  $5  at  one  time,  $7  at 
another,  and  $4  at  another.  He  afterward  drew  out  $9. 
How  much  was  left  in  the  bank  ? 

58.  A  cash-boy  earns  40  cents  a  day,  but  pays  15  cents 
for  lunch  and  5  cents  for  car  fare.  How  much  can  he  save 
each  day? 

59.  Soldahorse  that  cost  me  $100for  $90.  How  much 
did  I  lose  ? 

60.  From  a  field  containing  30  acres,  a  man  sold  5  acres 
to  one  man  and  10  acres  to  another.  How  many  acres  were 
left  in  the  field? 

61.  If  20  yards  of  cloth  are  cut  from  a  piece  containing 
50  yards,  how  many  yards  remain? 

62.  A  50-foot  pole  stands  5  feet  in  the  mud  and  10  feet 
in  the  water.     How  many  feet  are  above  the  water? 


26  SUBTRACTION. 

63.  A  bushel  contains  32  quarts,  and  a  peck  8  quarts. 
How  many  quarts  in  a  bushel  more  than  in  a  peck? 

64.  8  +  3  +  5  +  4-5  +  5  +  5  =  ? 

65.  8-5  +  3-5  +  5  +  10  +  3-8  =  ? 

66.  10  +  5  +  4-6  +  3-5  +  4-2  +  6-8  +  4-5  +  8  =  ? 

56.   Written. 
1.   From  824  subtract  567. 

Solution.  —  Write  the  subtrahend  under  the  minuend,  with  units 
of  the  same  order  under  a  vertical  line. 

Since  7  units  cannot  be  taken  from  4  units,  we  add  one  of 

824      the  tens,  which  equals  10  units,  to  the  4  units,  making  14 

567      units.     7  units  from  14  units  leaves  7  units,  which  we  place 

on  J      under  the  units'  column. 

Since  one  of  the  2  tens  was  added  to  the  units,  only  1 
ten  is  left.  6  tens  cannot  be  subtracted  from  1  ten,  so  one  of  the  9 
hundreds,  which  equals  10  tens,  is  added  to  the  1  ten,  making  11  tens. 
6  tens  from  11  tens  leaves  5  tens,  v^'hich  is  placed  under  the  tens'  column. 

Since  one  of  the  8  hundreds  was  added  to  the  tens,  only  7  hundreds 
are  left.  5  hundreds  from  7  hundreds  leaves  2  hundreds,  which  is 
placed  under  the  hundreds'  column. 

J?nie. — Place  the  subtrahend  binder  the  minuend,  with  units 
of  the  same  order  m  the  same  column. 
Beginning  at  the  right,  subtract  each  figure  of  the  subtrahend 
from  the  figure  of  the  minuend  directly  above  it,  and  write 
the  result  beneath. 
If  any  figure  in  the  minuend  is  less  than  the  corresponding 
figure  of  the  subtrahend,  increase  it  by  10  and  subtract. 
In  subtracting  the  next  column  diminish  by  1  the  figure 
in  the  miyiuend,  and  proceed  as  before. 

Proof. — Add,  the  subtrahend  and  reinainder.  If  the  result 
equals  the  minuend,  the  work  is  correct. 

Note.  — To  subtract  United  States  money,  write  the  numbers  with 
decimal  points  in  a  vertical  line.  Subtract  as  above,  and  place  the  point 
in  the  result  directly  under  the  points  above. 


WRITTEN   EXERCISES. 


27 


Subtract  and  prove : 


2. 

812          4.    315 

6.    210           8. 

1114        10.   416 

536                296 

156 

596               349 

3. 

521          5.   123 

7.    468          9. 

481          11.   811 

384                 98 

399 

319                 592 

12. 

2819-   674 

18. 

2763  - 1289 

24.   9327-8291 

13. 

8203  - 1276 

19. 

7284-4287 

25.   2840-1876 

14. 

4295-   597 

20. 

6801  -  5463 

26.   7075-3096 

15. 

7306  - 1807 

21. 

8003-   921 

27.   1911-892 

16. 

5962  -  4689 

22. 

2874  - 1392 

28.   743-158 

17. 

1914  - 1758 

23. 

4211  -  3216 

29.   916-619 

30. 

35146  - 18469 

35.   $112.11-^89.21 

31. 

11191-   6876 

36.   $5.294 -$4,878 

32. 

41362  -  30974 

37.   $2.516 -$1,918 

33. 

21323  - 19873 

38.   $28.316 -$16,955 

34. 

91111-78888 

39.   $31.116 -$17,583 

40. 

1894  +  2037  - 1746  +  1084. 

41. 

From  8000  subtract  5674. 

Solution.  — There  are  0  units,  0  tens,  and  0  hundreds  in  the  min- 
uend, therefore  1  of  the  8  thousands  must  be  changed  to  hundreds, 
J  QQ^Q     leaving  7  thousands ;  1  of  the  10  hundreds  must  be  changed 
8000      *^  tens,  leaving  9  hundreds ;  1  of  the  10  tens  must  be  changed 
5674      ^^  units,  leaving  9  tens.     The  1  ten  is  equal  to  10  units.     By 
none      this  change  the  minuend,  8000,  becomes  7  thousands,  9  hun- 
dreds, 9  tens,  and  10  units,  without  being  changed  in  value. 
From  this  changed  minuend  we  take  the  units  of  the  subtrahend, 
according  to  the  rule. 


28 


J 

StJBTRACTION. 

Subtract  and 

prove  : 

42.   6000 

46. 

16000 

50. 

$10.00 

3872 

11975 

8.756 

43.    90000 

47. 

28000 

51. 

25.00 

57624 

16042 

16.064 

44.   80000 

48. 

$1,005 

52. 

$34,572 

48324 

.876 

15.062 

45.   45000 

49. 

$28,004 

53. 

$10000.00 

28432 

17.555 

5873.168 

54.  From  seventeen  thousand  sixteen,  take  nine  thousand 
four  hundred  eighty-seven. 

55.  From    seventy -two  thousand  three  hundred  eleven, 
take  forty-six  thousand  nine  hundred  sixty-one. 

56.  Take  eight  thousand  four,  from  thirty  thousand. 

57.  The  minuend  is  7026  and  the  subtrahend  987.     Find 
the  difference. 

58.  Take  sixteen  thousand  eight  hundred  seventeen,  from 
twenty-four  thousand  five  hundred  forty-one. 

59.  The   difference   is   8037.       The   minuend   is   19406. 
Find  the  subtrahend. 

60.  Take  seventy-six  dollars  four  cents  two  mills,  from 
one  hundred  two  dollars  nine  mills. 

61.  From  sixty -five  thousand  three  hundred  sixteen,  take 
twenty-four  thousand  forty. 

62.  Take  $  86.215  from  $  900.09. 

63.  What  must  be  added  to  $  .67  to  make  $  3  ? 

64.  From  a  farm  of  263  acres  of  land  97  acres  were  sold. 
How  much  was  left  ? 

65.  A  man  had  $  279,  and  spent  $  129.64.     How  much 
had  he  left  ? 


WRITTEN   EXERCISES.  29 

66.  The  sum  of  two  numbers  is  4387,  and  the  smaller  is 
1925.     What  is  the  larger  number  ? 

67.  The  larger  of  two  numbers  is  2462,  and  their  difference 
is  537.     What  is  the  smaller  number  ? 

68.  A  man  sold  a  farm  for  f  10,250  which  cost  him  ^  7525. 
What  was  his  gain? 

69.  How  many  years  from  the  discovery  of  America, 
1492,  to  the  Declaration  of  Independence,  1776  ?  From  the 
settlement  of  St.  Augustine,  1565,  to  the  settlement  of  James- 
town, 1607  ? 

70.  A  man,  dying,  bequeathed  to  his  son  $  10,350,  and 
$  3475  less  to  his  daughter.  How  much  did  the  daughter 
receive  ? 

71.  In  a  school  of  946  pupils,  457  are  girls.  How  many 
are  boys  ? 

72.  The  population  of  the  United  States,  as  shown  by  the 
census  of  1900,  was  76,295,220.  The  census  of  1890  showed 
a  population  of  63,006,000.  What  was  the  increase  in  the 
ten  years  ? 

73.  How  much  higher  is  Mt.  Logan  than  Mt.  Mitchell, 
the  former  being  19,500  feet,  and  the  latter  6711  feet  high  ? 

74.  On  July  1,  1866,  the  public  debt  of  the  United 
States  was  $2,773,236,174,  and  on  December  1,  1891, 
$  1,546,961,696.     What  was  the  amount  of  reduction  ? 

75.  How  much  farther  than  Chicago  is  Omaha  from 
Boston,  Chicago  being  1117  miles  and  Omaha  1600  miles 
from  Boston  ? 

76.  How  much  larger  is  Iowa  than  New  York,  Iowa  hav- 
ing an  area  of  56,025  square  miles,  and  New  York,  49,170 
square  miles  ? 

77.  At  an  election  A  received  1524  v^Dtes  less  than  B,  who 
received  14,632  votes.     How  many  votes  did  A  receive? 


30  SUBTRACTION. 

78.  The  population  of  London  in  1891  was  4,211,060,  and 
of  Paris  in  the  same  year  2,447,960.  How  many  more 
inhabitants  had  London  than  Paris  ? 

79.  A  man  wishing  to  make  a  journey  of  482  miles  in 
three  days,  went  145  miles  the  first  day,  and  162  miles  the 
second  day.  How  many  miles  must  he  travel  on  the  third 
day  to  complete  the  journey  ? 

80.  A  man,  dying,  left  an  estate  of  f  75,000,  of  which  he 
bequeathed  $  25,000  to  his  widow,  1 10,750  to  each  of  his 
two  daughters,  and  the  remainder  to  his  son.  How  much 
did  his  son  receive  ? 

81.  A,  B,  and  C  entered  into  a  partnership  with  a  joint 
capital  of  $  45,000,  of  which  A  invested  $  17,250,  B  $  22,500, 
and  C  the  remainder.     What  was  C's  investment  ? 

82.  Mr.  Smith  had  on  deposit  in  the  bank  $  6125.56,  but 
drew  out  at  one  time  $  1080,  at  another  time  $  764.29,  and 
at  another  $  2150.27.     What  remained  in  bank  ? 

83.  A  man  bought  a  city  lot  for  $  2575,  and  after  build- 
ing a  house  upon  it  for  ^  5325,  and  a  barn  for  $  1075,  he  sold 
it  for  $10,250.     What  was  his  gain  ? 

84.  Two  boys  start  3000  yards  apart-  and  walk  toward 
each  other.  How  far  are  they  apart  when  one  has  walked 
1227  yards  and  the  other  932  yards  ? 

85.  The  sum  of  5423  and  3685  is  how  many  more  than 
the  difference  between  10,108  and  4345  ? 


MULTIPLICATION. 


57.  1.  If  one  orange  costs  3  cents,  what  will  4  oranges 
cost? 

2.  3  times  2  apples  are  how  many  apples  ? 

3.  If  there  are  3  trees  in  a  row,  how  many  trees  in  5 
rows  ? 

4.  What  will  5  tops  cost,  if  1  top  costs  5  cents  ? 

58.  Multiplication  is  the  process  of  finding  a  number  that 
is  a  given  number  of  times  another  number. 

59.  The  Multiplicand  is  the  number  multiplied. 

60.  The  Multiplier  is  the  number  multiplied  by. 

61.  The  result  of  multiplication  is  called  the  Product. 

62.  The  Factors  of  a  product  are  the  numbers  which,  mul- 
tiplied together,  will  produce  it. 

63.  The  Sign  of  Multiplication  is  an  oblique  cross,  x .  It 
means  multiplied  by  or  times. 

Thus  8x4  may  be  read  8  multijjlied  by  4,  when  the  first 
is  the  multiplicand,  and  8  times  4  when  the  first  is  the  mul- 
tiplier. 

Principles.  —  The  multiplier  must  be  an  Abstract  Number. 
The  multiplicand  and  product  must  be  like  numbers. 
When  both  factors  are  abstract,  either  may  be  taken  as  the 
multiplier  or  multiplicand. 

31 


32 


MULTIPLICATION. 

64.     MULTIPLICATION  TABLES. 


1x1  = 

1 

2x1=      2 

3x1=      3 

4x1=      4 

1x2  = 

2 

2x2=      4 

3x2=      6 

4x2=      8 

1  X    3  = 

3 

2x3=      6 

3x3=      9 

4  X    3  =    12 

1  X    4  = 

4 

2x4=      8 

3  X    4  =    12 

4  X    4  =    16 

1x5  = 

5 

2  X    5  =    10 

3  X    5=    15 

4  X    5  =    20 

1x6  = 

6 

2  X    6  =    12 

3  X    6  =    18 

4  X    6  =    24 

1x7  = 

7 

2  X    7  =    14 

3  X    7  =    21 

4  X    7  =    2a^ 

1  X    8  = 

8 

2  X    8  =    16 

3  X    8  =    24 

4  X    8  =    32 

1x9  = 

9 

2  X    9=    18 

3  X    9  =    27 

4  X    9  =    36 

1  X  10  = 

10 

2  X  10  =    20 

3  X  10  =    30 

4  X  10  =    40 

1  X  11  = 

11 

2  X  11  =    22 

3  X  11  =    33 

4  X  11  =    44 

1  X  12  = 

12 

2  X  12  =    24 

3  X  12  =    36 

4  X  12  =    48 

5x1  = 

5 

6x1=      6 

7x1=      7 

8x1=      8 

5x2  = 

10 

6  X    2  =    12 

7  X    2=    14 

8  X    2  =    16 

5x3  = 

15 

6  X    3  =    18 

7  X    3=    21 

8  X    3  =    24 

5x4  = 

20 

6  X    4  =    24 

7  X    4  =    28 

8  X    4  =    32^ 

5x5  = 

25 

6  X    5  =    30/ 

7  X    5  =    3^ 

8  X    5  =    40 

5x6  = 

30^ 

6  X    6  =    36 

7  X    6  =    42 

8  X    6  =    48 

5x7  = 

35 

6  X    7  =    42^ 

7  X    7  =    49 

8  X    7  =    56 

5x8  = 

40 

6  X    8  =    48 

7  X    8  =    56 

8  X    8  =    64 

5x9  = 

45 

6  X    9  =    54 

7  X    9  =    63 

8  X    9  =    72 

5  X  10  = 

50 

6  X  10  =    60 

7  X  10  =    70 

8  X  10  =    80 

5x11  = 

55 

6  X  11  =    66 

7  X  11=    77 

8  X  11  =    88 

5  X  12  = 

60 

6  X  12  =    72 

7  X  12  =    84 

8  X  12  =    96 

9x1  = 

9 

10  X    1  =    10 

11  X    1  =    11 

12  X    1  =    12 

9x2  = 

18 

10  X    2  =    20 

11  X    2  =    22^ 

12  X    2  =    24, 

9x3  = 

27. 

10  X    3  =    30 

11  X    3  =    33 

12  X    3  =    36 

9x4  = 

36 

10  X    4  =    40 

11  X    4  =    44 

12  X    4  =    48 

9x5  = 

45 

10  X    5  =    50 

11x5=    55 

12  X    5  =    60 

9x6  = 

54 

10  X    6  =    60 

11  X    6  =    66 

12  X    6  =    72 

9x7  = 

63 

10  X    7  =    70 

11  X    7=    77 

12  X    7  =    84 

9x8  = 

72 

10  X    8  =    80 

11  X    8  =    88 

12  X    8  =    96 

9x    9  = 

81 

10  X    9  =    90 

11  X    9  =    99 

12  X    9  =  108 

9  X  10  = 

90 

10  X  10  =  100 

11  X  10  =  110 

12  X  10  =  120 

9  X  11  = 

99 

10  X  11  =110 

11  X  11  =  121 

12  X  11  =  132 

9  X  12  = 

108 

10  X  12  =  120 
/ 

11  X  12  =  132 

12  X  12  =  144 

OKAL   EXERCISES.  33 

Oral. 

5.  At  10  cents  a  dozen,  what  will  8  dozen  bananas  cost  ? 

6.  How  many  wings  have  5  birds  ? 

7.  Rose  has  8  marbles,  and  her  brother  7  times  as  many. 
How  many  has  her  brother  ? 

8.  If  there  are  12  plants  in  5  rows,  how  many  plants 
in  all  ? 

9.  If   a  dozen  eggs  cost  12  cents,  what  will  10  dozen 
cost  ? 

10.  Eight  ten-cent  tablets  will  cost  how  much  ? 

11.  Harry  can  pick  4  quarts  of  berries  in  one  day.  At 
the  same  rate,  how  many  can  he  pick  in  11  days  ? 

12.  What  will  7  tables  cost  at  |?  8  each  ? 

13.  How  many  days  in  8  school  weeks  ? 

14.  There  are  8  quarts  in  a  peck.  How  many  quarts  in 
12  pecks? 

15.  A  horse  can  travel  8  miles  an  hour,  and  a  man  5  miles 
an  hour.  If  they  start  from  the  same  point  at  the  same 
time,  and  travel  in  the  same  direction,  how  far  from  the 
starting  point  will  the  horse  be  in  9  hours  ?  How  far  will 
the  man  have  travelled  ?     How  far  apart  will  they  be  ? 

16.  There  are  7  days  in  a  week.  How  many  days  in  12 
weeks  ? 

17.  If  a  pound  of  sugar  costs  5  cents,  what  will  be  the 
cost  of  11  pounds  ? 

18.  In  the  last  10  problems  what  number  must  be  con- 
sidered as  the  multiplier  ? 

19.  If  Laura  can  read  12  pages  in  one  day,  how  many 
pages  can  she  read  in  12  days  ? 

20.  At  4  cents  each,  what  will  a  dozen  lead  pencils  cost? 


34  MULTIPLICATION. 

21.  There  are  12  inches  in  a  foot.     How  many  inches  in 
a  ten-foot  pole  ? 

22.  At  $6  a  barrel,  what  will  be  the  cost  of  9  barrels  of 
flour? 

23.  How  many  corners  have  eleven  squares  ? 

24.  What  will  be  the  cost  of  4  yards  of  ribbon  at  10  cents 
a  yard  ? 

65.   Drill  on  the  multiplication  table. 
Tell  the  products  quickly : 


5x3 

8x  11 

4x6 

5x4 

8x7 

4x12 

12x10 

5x7 

4x6 

6x2 

6x11 

3x3 

6x8 

5x6 

10x5 

8x10 

5x4 

9x3 

5x9 

4x7 

9x10 

6x5 

8x5 

11  xll 

6x4 

.2x8 

7x7 

7x4 

12x11 

12x5 

6x12 

8x9 

6x5 

2x11 

5x9 

9x11 

10x9 

5x5 

4x12 

7x11 

6x6 

11x5 

8x8 

7x12 

12x12 

Written. 

25.    How  many  are  5  times  476  ? 

Solution.  —  We  first  write  the  multiplier  under  the  multiplicand, 
and  begin  at  the  right  to  multiply.    5  times  6  units  are  30  units,  which 

are  3  tens  and  0  units.  We  place  the  0 
Multiplicand  476  ^^der  the  units.  The  3  tens  are  to  be  added 
Multiplier  5       with  the  tens'  product.     5  times  7  tens  are 

Product  oooA      35  tens  +  the  3  tens  are  38  tens,  which  are 

3  hundreds  and  8  tens.  Place  the  8  tens 
under  the  tens,  the  3  hundreds  to  be  added  to  the  hundreds'  product. 
5  times  4  hundreds  are  20  hundreds  +  the  3  hundreds  are  23  hun- 
dreds, which  are  2  thousands  and  3  hundreds.  We  place  the  3  hun- 
dreds under  the  hundreds  and  the  2  thousands  in  the  thousands'  place. 
Product,  2380. 


WRITTEN  EXERCrSES.  35 

Find  the  products : 

26.  2  X  324  34.  3921  x  11  42.  3  x  567892 

27.  5  X  413  35.  2834  x  12  43.  12  x  461581 

28.  4  X  728  36.  1895  x  4  44.  4  x  196482 

29.  6  X  283  37.  2306  x  5  45.  11  x  135426 

30.  7  X  426  38.  5872  x  6  46.  5  x  876948 

31.  8  X  529  39.  4936  x  7  47.  10  x  583876 

32.  9  X  492  40.  2875  x  8  48.  6  x  426345 

33.  10  X  263  41.  5987  x  9  49.  9  x  123456 

Note.  —  In  multiplying  United  States  money  place  the  decimal 
point  in  the  product  as  many  places  to  the  left  as  it  is  in  the  multiplicand. 

50.  $1.26x4  54.  $12,985x8  58.  3  x  $1,875 

51.  $2.87x5  55.  $19,723x5  59.  5  x  $3,732 

52.  $3.94  X  6  56.  $18,285  x  9  60.  8  x  $1,436 

53.  $8.75x7  57.  $22,487x4  '    61.  10  x  $2,904 

62.  Find  the  cost  of  10  books,  when  one  book  costs 
$1,375. 

63.  If  I  earn  $12.75  in  one  week,  how  much  do  I  earn 
in  9  weeks  ? 

64.  What  will  be  the  cost  of  12  yards  of  carpeting  at 
$1.08  a  yard? 

65.  If  there  are  144  pens  in  a  box,  how  many  pens  in  11 
boxes  ? 

66.  How  many  bushels  of  apples  will  four  orchards  pro- 
duce, if  there  are  84  trees  in  each  orchard,  and  the  average 
yield  is  10  bushels  to  a  tree  ? 

67.  Find  the  product,  when  the  multiplicand  is  1872  and 
the  multiplier  12. 

68.  In  one  mile  there  are  5280  feet.  How  many  feet  in 
6  miles  ? 


36  MULTIPLICATION. 

69.  137  oranges  were  sold  at  4  cents  each,  from  a  box 
containing  225.  The  remainder  were  sold  at  5  cents  each. 
What  did  the  oranges  sell  for  ? 

70.  Frank  had  $30.18,  and  James  5  times  as  much. 
How  much  have  both  ? 

71.  A  square  mile  contains  640  acres.  How  many  acres 
in  11  square  miles  ? 

72.  The  average  cost  of  building  a  certain  railroad  was 
$2328  a  mile.     What  was  the  cost  of  8  miles  of  such  road  ? 

73.  In  a  long  ton  there  are  2240  pounds.  How  many 
pounds  in  12  long  tons  ? 

74.  In  a  week  there  are  7  days  of  24  hours  each.  How 
many  hours  in  8  weeks  ? 

75.  One  cord  of  wood  contains  128  cubic  feet.  How 
many  cubic  feet  in  7  cords  of  wood  ? 

76.  A  man  bought  at  a  sale  6  bicycles  at  $25.25  apiece. 
What  was  the  entire  cost  ? 

77.  When  flour  is  $6.25  a  barrel,  what  will  be  the  cost 
of  12  barrels  ? 

78.  There  are  8  quarts  in  a  peck  and  4  pecks  in  a  bushel. 
How  many  quarts  in  10  bushels  ? 

79.  In  a  square  foot  there  are  144  square  inches.  How 
many  square  inches  in  6  square  feet  ? 

80.  Six  loads  of  coal  averaged  3948  pounds  each.  How 
many  pounds  in  the  6  loads  ? 

81.  There  are  12  units  in  a  dozen  and  12  dozen  in  a 
gross.     How  many  units  in  9  gross  ? 

66.    To  multiply  by  lo,  lOO,  looo,  etc. 
Rule. — Annex  to  the  multiplicand  as  many  naughts  as  there 
are  in  the  multiplier.     If  the  multiplicand  contains  cents 
or  mills,  remove  the  decimal  point  as  many  places  to  the 
right  as  there  are  naughts  in  the  multiplier. 


WRITTEN   EXERCISES. 


37 


67.  Write  the  products  without  copying : 

1.  125  X  10        6.    28  X  100  11.    $.36  x  1000 

2.  $3.64  X  10    7.    36  x  100 

3.  598  X  10        8.    284  x  100 

4.  369  xlO        9.    $3.95  x  100 

5.  786  X  10      10.    $4.69  x  100 


12.  24  X  10000 

13.  $2.93x100000 

14.  1398  X  100000 

15.  28732  X  100000 


Note.  —  Pupils  should  be  drilled  on  work  of  this  kind  until  they 
can  give  results  at  sight. 

16.    Multiply  297  x  5000. 

297  Solution.  —  5000    is  5   times    1000.      We    therefore 

5000      multiply  the  297  first  by  5,  and  annex  three  naughts  to 
1,485,000     the  product. 


Written. 

17.  368  X  20 

18.  429  X  20 

19.  672  X  30 

20.  395x50 

21.  875  X  90 


22.  486  X  200 

23.  328  X  400 

24.  594  X  600 

25.  398  X  700 

26.  429  X  900 


27.  369  X  3000 

28.  426  X  5000 

29.  598  X  60000 

30.  324  X  800000 

31.  986  X  9000000 


32.    What  is  the  product  of  386  multiplied  by  124  ? 


First  Solution.  —  We  write  the  multiplier  under  the  multiplicand, 

with  their   right-hand  figures  in  a 
column. 

Multiplying  by  the  4,  we  have 
1544,  the  first  partial  product.  The 
2,  standing  in  tens'  place,  equals 
20.     Multiplying   by  20,    we    have 


386 
124 


1544,  Ist  partial  product. 

7720,  2d  partial  product. 
38600,  3d  partial  product. 
47864,  sum  of  partial  products.     4720,  the  2d  partial  product.     The 

1  standing  in  the  hundreds'  place 
equals  100.  Multiplying  by  100,  we  have  38600,  the  3d  partial  product. 
The  sum  of  the  partial  products,  47864,  is  the  entire  product. 


38 


MtTLTIPLICATION. 


Second  Solution.  —  The  naughts  at  the  right  of  the  2d  and  3d 


386 
124 

1544  1st  partial  product. 

772  2d  partial  product. 

.' ) 8(5  3d  partial  product. 

47864,  sum  of  partial  products. 


partial  products  may  be  omitted, 
whereby  7720  units  become  772  tens, 
and  38600  units  become  386  hun- 
dreds. Care  must  be  observed  that 
the  right-hand  figure  of  772  tens 
falls  under  the  tens,  and  that  the 
right-hand  figure  of  386  hundreds 
falls  under  the  hundreds. 


Rule.  —  Write  the  multiplier  under  the  multiplicand  with 
their  right-hand  figures  in  a  line. 
Begin  at  the  right,  and  multiply  by  each  figure  of  the  multi- 
plier successively,  placing  the  right-hand  figure  of  each 
partial  product  directly  under  the  figure  used  as  a  multi- 
plier, a7id  add  the  partial  products. 

Proof.  —  Multiply  the  midtiplier  by  the  multiplicand.     If 
results  are  the  same,  the  work  is  probably  correct. 

Note  1.  — If  either  factor  contains  cents  or  mills,  the  product  must 
have  as  many  places  at  the  right  of  a  decimal  point  as  that  factor. 
Note  2.  —  When   naught   occurs  in  the   multiplier,  pass  it,  and 


multiply  by  the  next  figure. 

33.     1692 

204 

6768 

3384 

345168 

Find  the  products : 

35. 

2346  X  38 

36. 

1983  X  206 

37. 

1694  X  75 

38. 

$16.52x103 

39. 

$  87.53  X  87 

34.     32165 

2004 

128660 

64330 

64458660 

40. 

2831  X  2006 

41. 

$  3.542  X  35 

42. 

2984  X  1673 

43. 

1659  X  98 

44. 

46982  X  1394 

WRITTEN   EXERCISES.  30 

45.  2872  X  78  62.  75063  x  2060 

46.  39832  X  1073  53.  3858  x  70 

47.  $  29.34  X  52  54.  83692  x  5000 

48.  87531  X  4006  55.  2987  x  90 

49.  2556  X  48     '  56.  37942  x  9000 

50.  39842  X  2354  57.  3942  x  87 

51.  3872  X  25  58.  28432  x  1001 

To  multiply  when  the  unit  figure  of  the  multiplier  is  i . 

59.    Multiply  1634  x  41. 

1634  X  41         Solution.  —  Multiplying  by  1   simply   repeats  the 

f*KQa  multiplicand.      We  therefore  multiply  by  only  the  4 

'     -  tens,  writing  the  first  figure  one  place  to  the  left.     Add 

DDyy4:  ^j^lg  product  to  the  multiplicand. 

60.  3256  X  21  64.  2811  x  71 

61.  4872  X  31  65.  1987  x  81 

62.  3854  X  51  66.  2984  x  111 

63.  2958  X  61  67.  2872  x  121 

To  multiply  when  the  tens*  figure  of  the  multiplier  is  i. 

68.    Multiply  2876  x  15. 

Solution.  — Multiply  by  the  5,  writing  the  product 

2867  X  15       one  place  to  the  right  of  the  multiplicand,  and  add  the 

14335  product  to  the  multiplicand.     It  is  evident  that  multi- 

43005  plying  hy  1  ten  is  the  same  as  multiplying  by  10  units, 

in  which  case  the  product  is  28670  units,  or  2867  tens. 

69.  3824  X  16  73.  2468  x  17 

70.  1968  X  18  74.  1694  x  19 

71.  1542  X  14  75.  3259  x  15 

72.  3241  X  13  76.  2879  x  16 

77.  At  $  21  a  ton,  what  is  the  cost  of  234  tons  of  hay  ? 

78.  At  15  cents  each,  what  is  the  cost  of  254  pineapples  ? 


40  MULTIPLICATIOK. 

79.  If  a  train  can  run  at  an  average  speed  of  51  miles  an 
hour,  how  far  can  it  run  in  146  hours  ? 

80.  A  wholesale  grocer  purchased  1356  dozen  eggs  at  15 
cents  a  dozen.     What  did  they  cost  him  ? 

81.  A  common  soldier   receives  $14   a   month.     What 
must  be  paid  to  3586  soldiers  for  6  months  ? 

82.  At  f  31  a  head,  what  must  be  paid  for  3162  cattle  ? 

83.  324  X  18  X  41  =  ?  286  x  31  x  13  =  ? 

84.  256  X  15  X  51  =  ?  316  x  16  x  61  =  ? 

85.  Find  the  cost  of  312  acres  of  land  at  $  63  an  acre. 

86.  There  are  1760  yards  in  one  mile.     How  many  yards 
in  16  miles  ? 

87.  If  one  acre  of  land  will  yield  23  bushels  of  wheat, 
how  many  bushels  will  230  acres  yield  at  the  same  rate  ? 

88.  There  are  5280  feet  in  a  mile.     How  many  feet  in 
21  miles  ? 

89.  What  will  be  the  weight  of  1000  barrels  of  sugar,  if 
they  average  324  pounds  each  ? 

90.  If  a  man  earns  $  165  a  month,  what  will  he  earn  in  a 
year  at  the  same  rate  ? 

91.  How  many  pounds  in  100  barrels  of  flour,  each  barrel 
weighing  196  pounds  ? 

92.  If  a  train  runs  42  miles  an  hour,  how  many  miles  will 
it  run  in  36  hours  ? 

93.  What  will  be  the  cost  of    50  village  lots  at  $  675 
each  ? 

94.  If  a  garrison  of  men  consume  1048  pounds  of  meat  in 
one  day,  how  many  pounds  will  they  consume  in  28  days  ? 

95.  What  will  126  barrels  of  flour  cost  at  $  5.25  a  barrel  ? 

96.  What  will  be  the  cost  of  500  head  of  cattle  at  $42  a 
head  ? 


WRITTEN   EXERCISES.  41 

97.  If  the  profits  from  one  acre  of  land  are  $  16.25,  what 
are  the  profits  from  128  acres  at  the  same  rate  ? 

98.  There  are  63360  inches  in  a  mile.    How  many  inches 
are  there  in  71  miles  ? 

99.  How  many  hours  in  a  year  of  365  days,  there  being 
24  hours  in  one  day  ? 

100.  There  are  640  acres  in  a  square  mile.  How  many 
acres  in  the  State  of  Illinois,  which  has  56650  square  miles? 

101.  If  it  costs  $  122.39  to  feed  a  soldier  one  year,  what 
will  it  cost  to  feed  a  garrison  of  8527  soldiers  for  the  same 
time  ? 

102.  A  farmer  raised  on  his  farm  245  bushels  of  corn, 
298  bushels  of  wheat,  and  216  bushels  of  barley.  He  sold 
the  corn  at  $  .63,  the  wheat  at  $  .92,  and  the  barley  at  $  .85 
a  bushel.     What  did  he  receive  for  the  whole  ? 

103.  A  man  exchanged  city  property  valued  at  $  12,375 
for  a  farm  of  175  acres  at  $  64.75  an  acre.  How  much 
money  ought  he  to  receive  in  addition  to  the  farm  ? 

104.  If  294  men  can  do  a  piece  of  work  in  37  days,  how 
long  will  it  take  1  man  to  do  it  ? 

105.  A  man  starts  to  walk  from  New  York  to  Chicago, 
a  distance  of  900  miles.  When  he  has  walked  at  the  rate 
of  28  miles  a  day  for  28  days,  how  far  is  he  from  Chicago. 

106.  Two  railway  trains  start  2100  miles  apart  and 
travel  toward  each  other,  one  going  45  miles  an  hour  and 
the  other  55  miles  an  hour.  How  far  are  they  apart  after 
19  hours  ? 

107.  How  many  pounds  in  237  car-loads  of  corn,  each  car 
containing  425  bushels,  and  each  bushel  weighing  60  pounds  ? 

108.  A  grocer  bought  10  casks  of  molasses,  each  contain- 
ing 42  gallons,  at  35  cents  a  gallon,  and  sold  the  same  at  a 
profit  of  8  cejits  a  gallon.  What  was  the  selling  price,  and 
jbhe  profit  ? 


DIVISION. 


68.  1.    A  boy  paid  8  cents  for  2  oranges.     What  did  he 
pay  for  one  orange  ? 

2.  At  4  cents  each,  how  many  oranges  can  be  bought  for 
12  cents  ? 

3.  A  man  divided  $  50  equally  among  his  5  children. 
How  much  did  each  receive? 

4.  How  many  groups  of  5  blocks  can  be  made  from  20 
blocks  ? 

5.  How  many  3's  in  9  ?     How  many  5's  in  15  ?     How 
many  4's  in  12  ? 

6.  How  many  times  5  is  30  ?    How  many  times  8  is  56  ? 

7.  There  are  3  feet  in  a  yard.     How  many  yards  in  a 
12-foot  pole  ? 

8.  A  boy  buys   apples  at  two  cents  apiece,  paying  14 
cents  for  them.     How  many  apples  does  he  buy  ? 

9.  A  boy  buys  7  apples  for  14  cents.     How  much  does 
he  pay  apiece  ? 

10.  Four  times  what  number  equals  12  ? 

11.  What  number  multiplied  by  5  equals  15. 

12.  16  is  4  times  what  number  ? 

69.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another. 

70.  The  number  divided  is  the  Dividend. 

42 


PRINCIPLES    OF   DI^n^li^^UFOjJjiife' 

71.  The  number  by  which  the  dividend  is  divided  is  the 
Divisor. 

72.  The  result  of  division  is  the  Quotient. 

73.  When  the  divisor  is  not  exactly  contained  in  the 
dividend,  the  part  of  the  dividend  that  is  left  is  the 
Remainder. 

74.  The  Sign  of  Division  is  -t-,  and  when  placed  between 
two  numbers  signifies  that  the  first  is  to  be  divided  by  the 
second. 

Thus,  56  -7-  8  is  read  56  divided  by  8. 

Division  is  also  indicated  by  writing  the  dividend  above 

the  divisor  with  a  line  between  them. 

56 
Thus,  —  may  be  read  56  divided  by  8. 
8 

75.  Principles. — 1.  The  remainder  and  dividend  are 
like  numbers. 

2.  When  the  divisor  is  abstract,  the  dividend  and  quotient 
are  like  numbers. 

3.  When  the  dividend  and  divisor  are  concrete,  the 
quotient  is  abstract. 

4.  The  product  of  the  divisor  and  quotient  plus  the 
remainder  equals  the  dividend. 

What  are  like  numbers  ? 
What  is  an  abstract  number  ? 
What  is  a  concrete  number  ? 

To  the  teacher. 

Write  a  problem  requiring  a  remainder  and  show  that  it 
is  like  the  dividend. 

Write  a  problem  requiring  the  divisor  to  be  an  abstract 
number. 

Write  a  problem  requiring  the  quotient  to  be  an  abstract 
number. 


44 


DIVISION. 


DIVISION  TABLES. 


1- 

-1=  1 

2- 

-  2=  1 

3- 

-  3=  1 

4-4=1 

2- 

-1=2 

4  - 

-2=2 

6  - 

-3=2 

8- 

-4=2 

3- 

-1=3 

6- 

-2=3 

9- 

-3=3 

12  - 

-4=3 

4- 

-1=4 

8  - 

-2=4 

12- 

-3=4 

16- 

-4=4 

5- 

-1=5 

10- 

-2=5 

15- 

-3=5 

20- 

-4=5 

6- 

-1=6 

12- 

-2=6 

18- 

-3=6 

24- 

-4=6 

7  - 

-  1  =  7 

14- 

-2=7 

21- 

-  3=  7 

28- 

-4=7 

8- 

-1=  8 

16- 

-2=8 

24- 

-  3=  8 

32- 

-4=8 

9- 

-1=9 

18- 

-2=9 

27- 

-3=9 

36- 

-4=9 

10- 

-1  =  10 

20- 

-2  =  10 

30- 

-  3  =  10 

40- 

-  4  =  10 

n  - 

-1=11 

22- 

-  2=  11 

33- 

-  3=  11 

44  - 

-  4  =  11 

12- 

-  1  =  12 

24- 

-  2  =  12 

36- 

-  3  =  12 

48- 

-  4  =  12 

5h-5=  1 

6- 

-  6=  1 

7- 

-7=1 

8- 

-8=1 

10-5=  2 

12- 

-  6=  2 

14  - 

-7=2 

16- 

-8=2 

15-5=  3 

18- 

-  6=  3 

21  - 

-7=3 

24- 

-8=3 

20-5=  4 

24- 

-  6=  4 

28- 

-  7=  4 

32- 

-8=4 

25-5=  5 

30- 

-  6=  5 

35- 

-7=5 

40- 

-8=5 

30-5=  6 

36- 

-  6=  6 

42- 

-7=6 

48- 

-8=6 

35-5=  7 

42- 

-  6=  7 

49- 

-7=7 

56 

-  8=  7 

40  -  5  =  8 

48- 

-6=8 

56- 

-  7=  8 

64- 

-8=8 

45-5=  9 

54- 

-  6=  9 

63- 

-  7=  9 

72- 

-  8=  9 

50  -  5  =  10 

60- 

-  6  =  10 

70- 

-  7=  10 

80- 

-  8  =  10 

55-5  =  11 

66- 

-6=11 

77- 

-  7  =  11 

88- 

-8  =  11 

60  -  5  =  12 

72- 

-  6=  12 

84- 

-  7  =  12 

96- 

-  8  =  12 

9-9=  1 

10- 

-10=  1 

11  - 

-11=  1 

12- 

-12=  1 

18-9=  2 

20- 

-10=  2 

22- 

-11  =  2 

24- 

-  12  =  2 

27  -  9  =  3 

30- 

-10=  3 

33- 

-11=  3 

36- 

-  12  =  3 

36-9=  4 

40- 

-  10=  4 

44- 

-11=  4 

48- 

-12  =  4 

45-9=  5 

50- 

-10=  5 

55- 

-11=  5 

60- 

-  12  =  5 

54-9=  6 

60- 

-10=  6 

66- 

-11=  6 

72- 

-12=  6 

63-9=  7 

70- 

-10=  7 

77  - 

-11=  7 

84- 

-12  =  7 

72-9=  8 

80- 

-  10=  8 

88- 

-11=  8 

96- 

-12=  8 

81  -  9  =  9 

90- 

-10=  9 

99- 

-11=  9 

108- 

-12=  9 

90  -  9  =  10 

100- 

-  10  =  10 

110- 

-  11  =  10 

120-4 

-  12  =  10 

99  -  9  =  11 

110- 

-10=  11 

121  - 

-  11  =  11 

132- 

-12  =  11 

108  -  9  =  12 

120- 

-  10  =  12 

132  -^ 

-  11  =  12 

144- 

-  12  =  12 

DRILL   IN    DIVISION. 


45 


DRILL  IN  DIVISION. 


76.    Tell  the  quotients  quickly 


¥ 

¥, 

¥. 

ih  ¥, 

¥.  ¥,  ¥/. 

36-9 

40- 

-    4 

18-9 

20-4 

120  - 10 

40-5 

.55- 

-    5 

24-8 

44-11 

63 -i-    7 

36-4 

54- 

-    6 

15-3 

•       22  ^  11 

96H-12 

45-5 

56- 

-    7 

16 -i- 4 

15-5 

84-12 

25-5 

64- 

-    8 

27  H- 9 

84-12 

108 --    9 

30-6 

81- 

-    9 

28-4 

54-=-    9 

120  -  12 

35-7 

120- 

-10 

60-5 

36-4 

132  - 12 

56-8 

132- 

-11 

72 --8 

45-9 

110  -5-  10 

72-9 

144- 

-12 

77^7 

100  - 10 

121  - 11 

77.   Oral. 

1.  At  $5  a  barrel   how  many  barrels  of  flour  can  be 
bought  for  $55? 

2.  A  cook  uses  9  eggs  a  day.     At  this  rate,  in  how  many 
days  will  she  use  108  eggs  ? 

3.  If  $  63  is  equally  divided  among  9  men,  how  many 
dollars  will  each  man  receive  ? 

4.  The  product  of  two  numbers  is  84.     One  of  the  num- 
bers is  7.     What  is  the  other  ? 

5.  A  clothing  merchant  made  a  profit  of  $  54  on  9  suits. 
What  was  his  average  profit  per  suit  ? 

6.  How  many  tons  of  coal  at  $  4  a  ton  can  be  bought  for 

$48? 

7.  If  a  boy  earns  $  132  a  year,  how  much  does  he  earn  iu 
a  month  ? 

8.  If  8  apples  cost  24  cents,  what  will  one  apple  cost  ? 
How  many  apples  can  be  purchased  for  36  cents  ? 


46  DIVISION. 

9.    If  5  tables  cost  $  40,  how  many  tables  can  be  bought 
for  $  96  ? 

10.  There  are  32  quarts  in  a  bushel,  and  8  quarts  in  a 
peck.     How  many  pecks  in  a  bushel  ? 

11.  There  are  144  units  in  a  gross  and  12  units  in  a 
dozen.     How  many  dozen  in  a  gross  ? 

12.  How  many  quarts  of  berries  at  10  cents  a  quart  can 
I  buy  for  $  120  ? 

13.  One  kind  of  ribbon  costs  10  cents  a  yard,  and  another 
8  cents  a  yard.  How  many  more  yards  of  the  cheaper 
ribbon  can  I  buy  for  80  cents,  than  of  the  dearer  ? 

14.  What  numbers  must  be  divided  by  8  to  get  the 
following  quotients  ?     3,  5,  7,  12,  6,  8,  4,  2,  11,  10. 

15.  Name  the  numbers  which  multiplied  by  7  will  give 
56,  63,  14,  21,  84,  77,  40,  24,  49,  42. 

16.  If  for  ^40  I  can  buy  8  pictures,  how  many  pictures 
can  I  buy  for  $60? 

17.  How  many  5  cent  pieces  in  50  cents  ? 

18.  Into  how  many  fields  of  12  acres  each  can  I  divide  a 
farm  containing  132  acres  ? 

19.  My  expenses  for  an  8-week  vacation  were  $56. 
What  were  my  average  expenses  per  week  ? 

20.  If  4  tons  of  coal  cost  $  16,  how  many  tons  can  be 
bought  for  $  36  ? 

21 .  A  florist  has  a  rectangular  bed  of  carnations  containing 
72  plants  placed  in  6  straight  rows.    How  many  in  each  row? 

22.  How  many  lO's  in  90  ?  8's  in  72  ?  7's  in  56?  5's 
in  45  ?     8's  in  96  ? 

23.  In  a  schoolroom  there  are  56  desks  in  7  equal  rows. 
How  many  desks  in  each  row  ? 

24.  Bananas  at  48  cents  a  dozen  will  cost  how  much 
apiece  ? 


ORAL   EXERCISES.  47 

78.  When  there  is  a  remainder  after  dividing,  it  may  be 
placed  after  the  quotient,  thus,  76  -j-  9  =  8,  and  4  remainder ; 
or  it  may  be  placed  above  the  divisor  with  a  line  between 
as  a  part  of  the  quotient. 

Thus,  76  ^  9  =  8f  That  is,  9  is  contained  in  76,  8f 
times. 

79.  Wheii  a  whole  thing  is  divided  into  2  equal  parts, 
each  part  is  called  one  half,  written  i ;  when  divided  into 
3  equal  parts,  each  part  is  called  one  third,  written  \ ;  when 
into  4  equal  parts,  one  fourth,  written  \  ;    etc. 

80.  One  or  more  of  the  equal  parts  of  a  whole  is  called  a 
Fraction. 

1.  Write  one  fifth,  2  thirds,  3  fourths,  five  sixths,  seven 
eighths. 

9       Imparl    257       9       1156 

To  find  ^  oi  a.  number  divide  it  by  2. 
To  find  \  oi  A  number  divide  it  by  3. 

3.  How  can  you  find  J  of  a  number ?     \?    i?    \?     J^? 

-A? 

4.  How  much  is  i  of  12  ?     J  of  15  ?     \  of  20  ? 

5.  If  8  barrels  of  flour  are  worth  $  48,  what  is  one  barrel 
worth  ? 

Solution.  —  Since  8  barrels  are  worth  $  48,  one  barrel  is  worth  \  of 
$  48,  or  $  6.     Ans. 

6.  W^hen  10  books  cost  ^  40,  what  is  the  price  of  one 
book.     Ans.  J^  of  $  40,  or  f  4. 

7.  A  paper-boy  makes  60  cents  in  5  days.  How  much 
does  he  make  in  a  day  ? 

8.  Eight  boys  share  equally  in  a  division  of  40  cents. 
What  is  each  boy's  share  ? 

9.  Four  boys  share  equally  in  eating  a  melon.  What 
part  of  it  does  each  have  ? 


48  DIVISION. 

Written. 

81.  1.  Divide  1635  by  5. 

Solution.  —  The  divisor  is  written  at  the  left  of  the  dividend,  and 
separated  from  it  by  a  curved  line. 

Beginning  at  the  left,  5  is  not  contained  in  1  thou- 

5)1635       sand,  so  we    change    1    thousand   to   hundreds  (=10 
327       hundreds)  and  add  it  to   the  6  hundreds,  making   16 
hundreds. 

5  is  contained  in  16  hundreds  3  hundreds  times,  with  a  remainder 
of  1  hundred.  Place  the  3  hundreds  under  the  hundreds  of  the  divi- 
dend, change  the  1  hundred  to  tens  and  add  the  3  tens  of  the  dividend, 
and  we  have  13  tens.  5  is  contained  in  13  tens  2  tens  times,  with  a 
remainder  of  3  tens.  Place  the  2  tens  under  the  tens  of  the  dividend, 
change  the  3  tens  to  units,  and  add  the  5  units  of  the  dividend,  and  we 
have  35  units.  5  is  contained  in  35  units  7  units  times.  Place  the.  7 
units  under  the  units  of  the  dividend. 

The  quotient  is  327. 

We  prove  the  work  by  multiplying  divisor  and  quotient  to  get  the 
dividend.     327  x  5  =  1635.     Therefore  the  work  is  correct. 

The  above  method  is  called  Short  Division,  and  is  used 
when  the  divisor  is  12  or  less. 

82.  Divide  by  short  division : 


2. 

3. 

4. 

5. 

5)38465 
7693 

6)26543 
4423f 

5)69840 
13968 

7)3169842 
452834* 

Note.  —  When  the  dividend  contains  cents  or  mills,  and  the  divisor 
is  abstract,  there  must  be  as  many  places  in  the  quotient  at  the  right 
of  a  decimal  point  as  in  the  dividend. 

6.  2555 --5  12.  2796-8 

7.  3928 -f- 6  13.  3879-3 

8.  5860-5  14.  2984-9 

9.  3604-4  15.  ^if^ 

10.  2890-7       16.  ^-^^ 

11.  3984-5       17.  ^U^ 


18. 

12358 

19. 

5^345 

20. 

6iL4_8a 

21. 

^8^ 

22. 

lA^Sil 

23. 

98461 

WRITTEN    EXERCISES.  49 

24.  $600,005^5  29.  $292.80    ^12 

25.  $288,012-5-12  30.  $365.50   -5-5  • 

26.  $  300.010 -=- 10  31.  $286.92    --9 

27.  $990.011 --11  32.  $333,333^11 

28.  $333,264-11 

33.  Divide  the  sum  of  369  and  381  by  their  difference. 

34.  How  many  barrels  of  apples  at  $5  a  barrel  can  be 
purchased  for  $2535? 

35.  Paid  for  6  tons  of  coal  $31.50.  What  was  the  price 
per  ton  ? 

36.  $96.64  was  divided  equally  among  8  men.  How 
much  did  each  man  receive  ? 

37.  How  much  is  i  of  $  68.20  ? 

38.  A  farmer  bought  8  cows  for  $  283.60.  What  was  the 
average  cost  of  each  cow  ? 

39.  If  the  dividend  is  369876  and  the  divisor  9,  what  is 
the  quotient  ?     The  remainder  ? 

40.  $39848  was  left  to  several  children.  A  received  J  of 
it,  and  B  ^.     What  was  A's  share  ?     B's  share  ? 

41.  The  product  of  two  factors  is  38838;  one  of  them  is 
9.     What  is  the  other  ? 

42.  A  man  earns  $1500  a  year.  His  son  earns  \  as 
much.     How  much  does  the  son  earn? 

43.  Two  cities  are  432  miles  apart.  How  fast  must  a 
train  run  to  make  the  distance  in  9  hours  ? 

44.  I  paid  $45.90  for  9  tons  of  coal.  What  did  I  pay 
per  ton  ? 

45.  A  farmer's  potato  crop  for  the  year  1900  amounted 
to  3875  bushels.  ^  of  them  were  destroyed  by  blight.  How 
many  bushels  were  destroyed  ? 


50 


DIVISION. 


46.  Frank  had  $  30.18,  Eose  ^  as  much,  and  James  twice 
as  much.     How  much  did  all  have  ? 

47.  When  6  dozen  lemons  cost  ^1.44,  what  will  4  dozen 
cost  ? 

48.  A  man  had  3820  head  of  cattle  and  sold  i  of  them. 
How  many  did  he  sell  ? 

83.   To  divide  by  lo,   loo,  looo,  etc. 

Rule.  —  Cut  off  from  the  right  of  the  dividend  as  many  figures 
as  there  are  naughts  at  the  right  of  the  divisor. 
If  the  dividend  contains  cents  or  mills,  remove  the  decimal 
point  as  mayiy  places  to  the  left  as  there  are  naughts  in 
the  divisor. 


1. 

1|00)369|00 
369 


2. 

1|00)283|25 


283t¥^ 


3. 


I|000)36i954 


'^a  9  54 


10)^38.50 
$3.85 


100)$  2836.402 
$  28.364 


6. 

1000)$  28604.25 
$  28.604 


Note.  —  More  than  three  figures  are  not  needed  at  the  right  of  the 
decimal  point  in  the  quotient. 


Divide : 

7.  38492  by  100 

8.  29648  by  100 

9.  16900  by  100 

10.  38700  by  100 

11.  28000  by  1000 

12.  39642  by  1000 


13.  28006    by  1000 

14.  198751   by  1000 

15.  $138.05  by  10 

16.  $289.04  by  100 

17.  $1398.20  by  1000 

18.  $  2984.06  by  100 


LONG  DIVISION.  51 


LONG  DIVISION. 

84.  1.   Divide  3653  by  24. 

Solution.  —  24  is  not  contained  in  3  thousands,  so  we  change  3 

thousands  to  hundreds,  and  add  to  it  6  hundreds,  making  30  hundreds. 

Ap'cy^  24  is  contained  in  36  liundreds,  1  hundred  times.     We 

2"T     write  the  quotient  1  hundred  over  the  hundreds  of  the 

J4)oooo  dividend,  and  multiply  the  divisor  by  it.     This  gives  a 

24  product  of  24  hundreds,  which  we  write  under  the  36 

125  hundreds  and  subtract ;  the  remainder  is  12  hundreds, 

120  to  which  we  unite  the  5  tens,  making  125  tens. 

^  24  is  contained  in  125  tens,  5  tens  times.     We  write 

.r.  the  quotient,  5  tens  over  the  tens  of  the  dividend  and 

— -         multiply  the  divisor  by  it.     This  gives  a  product  of  120 

tens,  which  we  place  under  the  125  tens  and  subtract. 

The  remainder  is  5  tens,  to  which  we  unite  the  3  units,  making  53 

units. 

24  is  contained  in  53  units,  2  units  times.  We  write  the  quotient 
2  units  over  the  units  of  the  dividend,  and  multiply  the  divisor  by  it. 
This  gives  a  product  of  48  units,  which  we  write  under  the  53  units 
and  subtract.  The  remainder  is  5,  which  we  write  over  the  divisor  as 
part  of  the  quotient. 

Hence  the  quotient  is  152^^. 

We  prove  by  multiplying  the  divisor,  24,  by  the  quotient,  152,  and 
adding  the  remainder  to  the  product.  The  sum  is  the  dividend,  which 
proves  the  work  to  be  correct. 

85.  Rule. —  Write  the  divisor  at  the  left  of  the  dividend, 
with  a  curved  line  between. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest 
figures  on  the  left  of  the  dividend,  and  write  the  result  over 
the  right-hand  figure  of  this  partial  dividend,  as  the  first 
quotient  figure. 

Multiply  the  divisor  by  this  quotient  figure,  subtract  the 
product  from  the  partial  dividend,  and  to  the  remainder 
tinite  the  next  figure  of  the  dividend,  for  the  secoyid  par- 
tial dividend. 

Proceed  as  before  until  all  the  figures  of  the  dividend  have 
been  united  to  the  remainder. 


52 


DIVISION. 


If  any  partial  divideyid  does  not  contain  the  divisor,  place  a 
naught  in  the  quotient,  bring  down  the  next  figure  of  the 
dividend,  and  proceed  as  before. 

If  there  is  a  remainder  after  dividing  the  last  partial  divi- 
dend, write  it  over  the  divisor  at  the  right  of  the  quotient. 

Proof.  —  Find  the  product  of  the  divisor  and  quotient,  add 
the  remainder,  if  any,  and  if  the  sum  equals  the  dividend,  the 
work  will  be  right. 

"Find  the  quotients : 


2.  2578  -!- 16 

3.  11366 --37 

4.  10872-18 

5.  12572 --39 

6.  15966 --42 

7.  12096 --25 

8.  19436  H- 47 

23.  $96.64 --16 

24.  $137.34 --18 

25.  $62,826-74 

26.  $  2098.119 -- 987 

27.  $  671178.90 -f- 98 

28.  $  205877.75 -r- 25 

29.  $375561.50-50 

30.  $  104204.112 -- 40 


9.  28059 --36 

10.  31583 --46 

11.  24109-51 

12.  32695-57 

13.  33874-5-49 

14.  99003-^25 

15.  45914 --59 


16.  335630 -f- 62 

17.  491289-^73 

18.  216428^84 

19.  412582 -f- 58 

20.  981384-75 

21.  912946-24 

22.  427473 --97 

31.  29067642-1032 

32.  65980064-5004 

33.  27905138 -- 2116 

34.  796529184-3052 

35.  98053273-1350  ' 

36.  16119080 -- 2165 

37.  39849972-1960 

38.  89796438-9487 


39.  If  a  carriage  costs  $  74,  how  many  such  carriages  can 
be  purchased  for  $  62826  ? 

40.  What  number  multiplied  by  351  will  give  a  product 
of  347,692  ? 

41.  In  a  dollar  there  are  1000  mills.     How  many  dollars 
in  968,000  mills  ? 


WRITTEN   EXERCISES.  53 

42.  There  are  8  quarts  in  a  peck.  How  many  pecks  are 
there  in  1032  quarts  ? 

43.  How  many  bushels  are  there  in  24064  quarts,  there 
being  32  quarts  in  a  bushel  ? 

44.  ^ow  many  barrels  in  8232  pounds  of  flour,  there 
being  196  pounds  in  a  barrel  ? 

45.  If  165  acres  of  land  yield  5280  bushels  of  wheat, 
what  is  the  average  yield  for  an  acre  ? 

46.  If  a  railway  train  travels  at  the  rate  of  54  miles  an 
hour,  how  many  hours  will  it  require  to  travel  1944  miles? 

47.  The  product  of  two  numbers  is  72924,  and  one  of  the 
numbers  is  354.     What  is  the  other  number  ? 

48.  A  man,  dying,  left  property  to  the  amount  of  $  425000, 
of  which  he  gave  $  28350  for  a  public  library,  three  times 
as  much  to  charitable  institutions,  and  the  remainder  he 
divided  equally  among  100  persons.  How  many  dollars  did 
each  person  receive? 

49.  I  bought  27  tons  of  coal  for  $  148.50.  What  was  the 
price  of  a  ton  ? 

50.  A  landlord  has  provisions  sufficient  to  last  1  person 
391  days.     How  long  would  they  last  17  persons  ? 

51.  What  is  the  value  of  a  single  share  of  bank  stock  if 
500  shares  are  worth  $  63500  ? 

52.  There  are  1728  cubic  inches  in  one  cubic  foot.  How 
many  cubic  feet  are  there  in  93312  cubic  inches  ? 

53.  Which  is  cheaper,  and  how  much  per  yard,  thirteen 
yards  of  cloth  for  $  48.75,  or  65  yards  for  $  211.25? 

54.  A  drover  sold  135  head  of  cattle  at  $38  a  head,  and 
with  the  proceeds  bought  horses  at  $114  a  head.  How 
many  horses  did  he  buy  ? 


54  DIVISION. 

55.  At  what  rate  per  hour  should  a  railway  train  run  in 
order  to  travel  1944  miles  in  36  hours  ? 

56.  If  I  save  out  of  my  salary  $432  each  year,  in 
how  many  years  can  I  save  enough  to  buy  a  home  for 
$  6480  ? 

57.  The  dividend  is  287609,  the  quotient  904,  and  the 
remainder  137.     What  is  the  divisor  ? 

58.  A  farmer  has  fodder  enough  to  last  1  cow  224  days. 
How  many  days  will  the  same  fodder  last  9  cows  and  28 
sheep,  if  4  sheep  eat  as  much  as  1  cow  ? 

59.  If  it  costs  $  9687.50  a  year  to  board  62  persons,  how 
much  will  it  cost  to  board  1  person  for  the  same  time  ? 

60.  A  grocer  buys  520  barrels  of  flour  at  f  5.25  a  barrel. 
If  100  barrels  become  damaged  and  worthless,  for  how 
much  a  barrel  must  he  sell  the  remainder  in  order  to  get 
back  the  cost? 

61.  A  merchant  buys  324  yards  of  cloth  at  $  2.25  a  yard. 
For  how  much  per  yard  must  he  sell  it  in  order  to  gain 
$165.24? 

62.  Which  costs  the  more  per  bushel,  228  bushels  of 
wheat  for  $  207.48  or  12  bushels  for  $  10.80  ? 

63.  The  product  of  three  numbers  is  43200.  Two  of 
them  are  32  and  50.     What  is  the  third  ? 


INDICATED   OPERATIONS. 

86.  The  Parenthesis,  (  ),  indicates  that  all  the  numbers 
contained  therein  are  to  be  taken  together. 

87.  Brackets,  [  ],   Braces,   J  },  and  the  Vinculum, 


have  the  same  use  as  the  parenthesis.     These  are  called 
Signs  of  Aggregation. 


INDICATED   OPERATIONS.  65 

88.  When  the  parenthesis  is  not  used,  operations  indi- 
cated by  X  or  -j-  must  be  performed  first.     Thus, 

1.  12-r-4x2+36-f-4-2x4  =  ? 

Solution.  — 

12  -  4  X  2  =  6. 

36  -:-  4  =  9.  6  +  9-8  =  7.     Ans. 

2x4  =  8. 

2.  4  +  3x2  =  ?  5.    4x(3  +  2)  =  ? 

3.  (4  +  3)  x2  =  ?  6.    8  +  4-2  =  ? 

4.  4x3  +  2=?  7.    (8 +  4) -2  =  ? 

Note.  —  When  one  of  the  signs  of  aggregation  includes  another, 
the  operations  indicated  within  the  one  included  should  be  performed 
first,  and  the  sign  removed. 

89.  Find  the  value  of : 

1.  15  +  3  X6  +  10--5. 

2.  (6  +  4)  X  (3 +  2) -(8x5). 

3.  18H-3x2  +  8x2-r-4-6. 

4.  2 +  12 --4 -(10 +  6 -J- 4) --3. 
6.  11 +4-3  +  6x4. 

6.  3  +  4  X  6  -  (15  +  9  -  3). 

7.  164  +  16  -  250  -  10  +  16  X  3. 

8.  17 +  3x4x6  +  3-3  + 3. 

9.  [39  +  8  -  2  +  7]  X  6. 

10.    [6  +  15  X  3  -  6  +  16-8  +  4]  -8  +  5. 

PRINCIPLES   OF  DIVISION. 
Dividend.    Divisor.    Quotient. 

90.  24     -    6      =      4 

If  I  divide  the  dividend  by  tivice  the  divisor,  or  12,  the 
quotient  will  be  how  many  times  smaller  ?     24  —  (2  x  6)=  ? 

If  I  divide  twice  the  dividend,  or  48,  by  the  divisor,  the 
quotient  will  be  how  many  times  as  great  ?     (2  x  24)  —  6  =  ? 


66  INDICATED   OPERATIONS. 

If  I  divide  twice  the  dividend  by  twice  tlie  divisor,  how 
will  it  affect  the  quotient  ? 

If  I  divide  the  dividend  by  half  the  divisor,  or  3,  the  quo- 
tient will  be  how  many  times  as  large  ?     24  -^  (i-  of  6)  =  ? 

If  I  divide  half  the  dividend  by  the  divisor,  the  quotient 
will  be  how  many  times  smaller  ?     (i  of  24)  -j-  6  =  ? 

If  I  divide  halfth.Q  dividend  by  half  tliQ  divisor,  how  will 
it  affect  the  quotient  ?     (i  of  24)  --  (^  of  6)  =  ? 

From  the  above,  we  see  that  if  we  multiply  the  divisor, 
or  divide  the  dividend,  by  2,  we  divide  the  quotient  by  2. 

24  -  (2  X  6)  =  i  of  4,  or  2. 
(i  of  24)  -  6  =  i  of  4,  or  2. 

Also,  that  if  we  multiply  the  dividend  or  divide  the 
divisor  by  2,  we  multiply  the  quotient  by  2. 

(2  X  24)  --  6  =  2  times  4,  or  8. 
24  -J-  (i  of  6)  =  2  times  4,  or  8. 

Also,  that  if  we  multiply  or  divide  both  dividend  and 
divisor  by  2,  the  quotieyit  is  unchanged. 

(2  X  24)  -  (2  X  6)  =  4. 
a  of  24) -(1  of  6)  =  4. 

Hence,  the  following  principles  : 

Principles.  —  1.  Multiplying  the  divisor  or  dividing 
the  dividend  by  any  number  divides  the  quotient  by  that 
number. 

2.  Dividing  the  divisor  or  multiplying  the  dividend  by 
any  number  multiplies  the  quotient  by  that  number. 

3.  Multiplying  or  dividing  both  divisor  and  dividend  by 
the  same  number  does  not  affect  the  quotient. 


ORAL  REVIEW.  57 

MISCELLANEOUS  REVIEW. 
Oral. 

91.    1.    William   had   24   marbles,  and  lost   \  of  them. 
How  many  did  he  lose  ?     How  many  were  left  ? 

2.  How  many  quarts  are  in  J  of  a  bushel?  There  are 
32  quarts  in  a  bushel. 

3.  A  boy  found  50  cents.  He  kept  20  cents,  and  divided 
the  remainder  equally  between  his  two  sisters.  How  much 
did  each  receive  ? 

4.  After  paying  me  ^  40,  a  man  still  owes  me  $  60. 
How  much  did  he  owe  me  at  first  ? 

5.  If  5  oranges  cost  15  cents,  what  will  8  oranges  cost? 

Note.  — Solve  by  analysis. ' 

Thus,  since  5  oranges  cost  15  cents, 

1  orange  will  cost  \  of  15  cents,  or  3  cents, 
and  8  oranges  will  cost  8  times  3  cents,  or  24  cents. 
First  find  the  cost  of  1  orange,  then  the  cost  of  8  oranges. 

6.  H  12  pencils  cost  48  cents,  what  will  7  pencils  cost  ? 

7.  5  tons  of  coal  cost  $  25.  At  the  same  rate,  what  will 
be  the  cost  of  9  tons  ? 

8.  A  lady  bought  soda  for  10  cents,  sugar  for  12  cents, 
thread  for  5  cents,  ribbon  for  8  cents,  and  a  book  for  50 
cents.  How  much  change  should  she  receive  from  a  $1 
bill? 

9.  A  man  bought  a  horse  for  $  80,  and  paid  $  20  down. 
How  long  will  it  take  him  to  pay  the  balance  at  $5  a 
month  ? 

10.  Will  rode  40  miles  on  his  wheel,  and  Harry  20  miles 
more  than  that.     How  far  did  both  ride  ? 

11.  From  a  50-dollar  bill  was  spent  $5,  f  10,  $15,  and 
$  6.     How  much  remained  ? 


68,  MISCELLANEOUS    REVIEW. 

12.  A  farmer  had  40  chickens.  He  sold  -^  of  them,  and 
5  died.     How  many  were  left  ? 

13.  40  -  6  X  6  =  ?     (40  -  6)  X  6  =  ? 

14.  If  four  dresses  of  10  yards  each  are  cut  from  a  piece 
of  cloth  containing  80  yards,  how  many  yards  are  left  ? 

15.  John  and  James  travel  west;  John  going  at  the  rate 
of  8  miles  an  hour,  and  James  at  the  rate  of  5  miles  an 
hour.     How  far  apart  will  they  be  in  6  hours  ? 

16.  John  travels  east,  and  James  west,  each  making  5 
miles  an  hour.  How  far  apart  will  they  be  at  the  end  of 
the  first  hour  ?     The  second  hour  ?     The  sixth  hour  ? 

17.  If  John  travels  east  at  the  rate  of  8  miles  an  hour, 
and  James  west  at  the  rate  of  5  miles  an  hour,  how  far 
apart  will  they  be  at  the  end  of  6  hours  ? 

18.  If  6  men  do  a  piece  of  work  in  12  days,  in  what  time 
can  1  man  do  it  ? 

Analysis. — Since  6  men  do  the  work  in  12  days,  it  will  take  1 
man  6  times  as  long,  or  72  days. 

19.  If  4  men  can  dig  a  trench  in  9  days,  in  how  long  a 
time  can  6  men  dig  it  ? 

Analysis.  —  Since  4  men  dig  it  in  9  days, 

1  man  can  dig  it  in  4  x  9  days,  or  36  days, 
and  6  men  can  dig  it  in  ^  of  36  days,  or  6  days. 

20.  If  6  apples  cost  12  cents,  what  will  40  apples  cost  ? 

Written. 

21.  A  farmer  can  purchase  20  cows  for  $600.  At  this 
rate  what  will  8  cows  cost  ? 

22.  Two  trains  leave  Chicago,  one  going  east  and  the 
other  west.  One  runs  50  miles  an  hour  and  the  other  45 
miles.     How  far  apart  are  they  after  12  hours  ? 

23.  If  20  men  can  build  a  fence  in  15  days,  in  how  many 
days  could  50  men  build  the  same  fence  ? 


MISCELLANEOUS    KEVIEW.  69 

24.  Two  railway  trains  travel  toward  each  other  from 
cities  1000  miles  apart.  One  makes  55  miles  an  hour,  the 
other  45.     How  many  hours  will  elapse  before  they  meet  ? 

25.  If  354  tons  of  coal  cost  $  1770,  what  will  be  the  cost 
of  648  tons  ? 

26.  A  man  bought  land  for  $  8954  and  sold  it  for  $  12362, 
gaining  thereby  $  12  an  acre.    How  many  acres  were  there  ? 

27.  A  merchant  sold  324  yards  of  cloth  at  $2.76  a  yard, 
thereby  gaining  $  165.24.     What  did  it  cost  a  yard  ? 

28.  John  is  28  years  younger  than  his  father,  who  is  40 
years  old.  When  John  is  3  times  as  old  as  he  now  is,  how 
old  will  his  father  be  ? 

29.  Two  vessels  start  together,  going  in  the  same  direc- 
tion, one  at  the  rate  of  23  miles  an  hour,  and  the  other  36 
miles  an  hour.  In  how  many  hours  will  they  be  312  miles 
apart  ?  How  many  hours  would  be  required  if  the  vessels 
travel  in  opposite  directions  ? 

30.  In  gathering  huts,  Charles  gathered  354  quarts,  and 
Henry  286  quarts.  After  saving  64  quarts  apiece  for  them- 
selves, they  sold  the  remainder  for  $  33.60.  How  much  per 
bushel  did  they  receive,  there  being  32  quarts  in  a  bushel  ? 

31.  A  lady  finds  dress  goods  at  the  store  for  $1.25  and 
$  1.70  a  yard.  How  much  more  will  it  cost  her  to  buy  17 
yards  at  the  latter  price  than  the  same  amount  at  the 
former  ? 

32.  A  farmer  raised  255  bushels  of  wheat  and  376  bush- 
els of  potatoes.  He  sold  ^  of  the  wheat  at  $  .92  a  bushel 
and  J  of  the  potatoes  at  $  .40  a  bushel,  and  with  the  pro- 
ceeds bought  10  tons  of  hay.  How  much  was  the  hay  a 
ton? 

33.  If  I  buy  a  house  and  lot  for  $  6250,  paying  $  1375 
cash,  and  the  remainder  in  yearly  payments  of  $  325,  how 
many  years  will  be  required  to  pay  for  it  ? 


60  MISCELLANEOUS   REVIEW. 

34.  If  142  men  can  grade  a  street  in  36  days,  how  long 
will  it  take  72  men  to  do  it  ? 

35.  A  grocer  bought  12  barrels  of  sugar,  each  containing 
320  pounds,  at  5^  cents  a  pound,  and  5  chests  of  tea,  con- 
taining 52  pounds  each,  at  34  cents  a  pound.  He  gave  in 
payment  $  300.     How  much  change  did  he  receive  ? 

36.  A  water  tank  with  a  capacity  of  1918  gallons  has  run- 
ning into  it  a  pipe  with  a  flowing  capacity  of  319  gallons  an 
hour,  and  one  at  the  bottom,  leading  out  of  it,  that  can  dis- 
charge 456  gallons  an  hour.  If  the  tank  is  full  of  water  and 
both  pipes  running  at  their  full  capacity,  in  how  many 
hours  will  it  be  emptied? 

37.  A  man  bought  246  cows  at  f  38  a  head,  and  sold  them 
at  ^  45  a  head.     What  was  his  gain  ? 

38.  How  many  more  times  will  a  wheel  12  feet  in  cir- 
cumference turn  around  in  going  a  mile,  or  5280  feet,  than 
one  15  feet  in  circumference  ? 

39.  A  speculator  bought  324  pounds  of  butter  of  one  man 
and  295  pounds  of  another,  paying  23  cents  a  pound  in  each 
case.  He  sold  from  it  452  pounds  at  25  cents  a  pound. 
For  how  much  per  pound  must  he  sell  the  remainder  to  gain 
$  17.39  on  the  whole  by  both  sales  ? 

40.  Four  men.  A,  B,  C,  and  D,  together  bought  a  factory 
for  f  41765.  A  paid  $  10624,  B,  $  7156,  C,  $  780  less  than 
both  A  and  B,  and  D  the  remainder.  How  much  did  C  and 
Dpay? 

41.  I  bought  4  horses  at  one  time  for  $564,  6  horses  at 
another  time  for  f  852,  and  9  at  another  for  $  1548.  What 
was  the  average  price  paid  ? 


FACTOES. 


92.  The  Factors  of  a  number  are  the  integers  which  being 
multiplied  together  produce  the  number. 

Thus,  6  and  3  are  the  factors  of  18.  5,  2,  and  3  are  the 
factors  of  30. 

1.  Name  one  of  the  factors  of  16.  Name  three  factors  of 
16. 

2.  What  are  the  factors  of  24,  20,  36,  48,  60,  21,  44,  56, 
63,  50,  81,  64  ? 

3.  What  numbers  will  exactly  divide  54,  48,  63,  24,  35, 
84,  77,  96  ? 

93.  A  number  that  has  other  factors  besides  itself  and  1 
is  a  Composite  Number.  6,  8,  15,  are  composite  numbers.  A 
composite  number  is  divisible  hj  any  of  its  factors.  Its 
factors  are,  therefore,  exact  divisors  of  it. 

4.  Name  three  composite  numbers  and  their  factors. 
Name  three  numbers  that  are  not  composite. 

94.  A  number  that  has  no  factors  except  itself  and  1  is  a 
Prime  Number.     5,  7,  11,  17,  are  prime  numbers. 

5.  Name  three  prime  numbers.  Name  all  the  prime  num- 
bers from  1  to  30,  ?  ^^^    n     >  m  ^  >  -^  li  ^  ^1 

95    A  prime  number  used  as  a  factor  is  a  Prime  Factor. 

6.  3  and  5  are  prime  factors  of  15,  but  4  and  6  are  not 
prime  factors  of  24.     Why  not  ? 

Note. — Every  number  may  be  multiplied  by  1  to  produce  itself. 
While  both  are  factors,  according  to  the  definition,  they  are  not  so 
used  in  practice. 

61 


62  FACTORS. 

7.  Name  the  prime  factors  of  20,  18,  24,  25,  27,  38,  34, 
48,  49,  63. 

96.  A  number  that  is  divisible  by  2  is  an  Even  Number. 
Thus,  8,  6,  10,  14,  are  even  numbers. 

8.  Name  all  the  even  numbers  from  1  to  40. 

97.  A  number  not  divisible  by  2  is  an  Odd  Number. 
Thus,  3,  5,  9,  11,  are  odd  numbers. 

9.  Name  the  odd  numbers  from  1  to  40. 

98.  Every  prime  number  except  2  and  5  ends  with  1,  3, 
7,  or  9. 

10.  Name  a  prime  number  ending  with  1,  with  3,  with  7, 
with  9. 

11.  Name  a  composite  number  ending  with  1,  with  3, 
with  7,  with  9. 

2  is  an  exact  divisor  of  any  even  number,  or  of  any  num- 
ber ending  with  an  even  number,  or  with  0. 

3  is  an  exact  divisor  of  any  number  the  sum  of  whose 
digits  is  divisible  by  3.  4  is  an  exact  divisor  of  any  num- 
ber when  the  number  expressed  by  its  two  right-hand 
figures  is  divisible  by  4. 

5  is  an  exact  divisor  of  any  number  ending  with  0  or  5. 
Separating  a  number  into  its  factors  is  called  Factoring. 

99.  Written. 

12.  Find  the  prime  factors  of  60. 

^  Lr2:  Solution.  —  We  first  divide  by  tlie  prime  number  2. 

2  30  The  quotient,  30,  being  even,  we  also  divide  by  2.     The 

3  15  quotient  is  15,  which  we  divide  by  3,  giving  a  quotient 
5    5  of  5,  which  we  divide  by  5.     The  last  quotient  is  1. 

-j  Hence  the  prime  factors  of  60  are  2,  2,  8,  5. 

Rule. — Divide  the  given  niimher  by  cpiy  prime  number  that 
will  exactly  divide  it.  Divide  this  quotient  by  any  prime 
number,  and  so  continue  until  the  quotient  is  1.  The  several 
divisors  are  the  prime  factors. 


WRITTEN   EXERCISES.  63 

Find  the  prime  factors  : 

13.  63      18.  720  23.  2431  28.  13104 

14.  84      19.  1572  24.  2310  29.  11550 

15.  250     20.  1872  25.  7007  30.  17325 

16.  210     21.  2800  26.  3150  31.  64384 

17.  636     22.  2310  27.  3465  32.  10323 

CANCELLATION 

33.  What  is  the  quotient  of  42  -^  21  ? 

34.  What  is  the  quotient  of  3  x  2  x  7  divided  by  3  x  7  ? 

35.  How  many  times  is  3  times  5  contained  in  6  times 
5  ?     3  times  8  in  12  times  8  ? 

36.  Divide  12  x  16  by  4  x  16.     9  x  11  by  3  x  11. 

37.  Divide  18  x  7  by  9  x  7. 

38.  In  the  last  example  what  factor  is  found  in  both 
dividend  and  divisor  ? 

39.  Would  the  quotient  be  the  same  if  the  factor  7  were 
rejected  from  both  dividend  and  divisor  ? 

100.  Principles. — 1.  Eejecting  a  factor  from  a  number 
divides  the  number  by  that  factor. 

2.  Eejecting  the  same  factors  from  both  dividend  and 
divisor  does  not  affect  the  quotient.  (See  Principles  of 
Division.) 

101.  Rejecting  the  same  factors  from  both  dividend  and 
divisor  is  called  Cancellation. 

102.  1.    Divide  8  X  18  X  15  X  7  by  4  X  6  X  11  X  9. 

Solution.  —  We  first  indicate  the  division  by  writing  the  dividend 

over  the  divisor  with  a  line  between.     Since  4 

2^5  and  6  are  factors  of  8  and  18,  respectively,  they 

p  X  ]Lp  X  Jp  X  i       may  be  omitted,  or  cancelled,  from  both  divi- 

^  X  ^  X  11  X  f^       dend  and  divisor.     Since  8  in  the  dividend  is 

15       a  factor  of  9  in  the  divisor,  it  is  cancelled  from 


64  CANCELLATION. 

both,  leaving  3  in  the  divisor.     3  in  the  divisor,  being  a  factor  of  15 
in  the  dividend,  is  cancelled  from  both. 

The  product  of  the  uncancelled  factors  in  the  dividend  is  70,  and 
in  the  divisor  11.     The  quotient  is  therefore  |^,  which  equals  6^. 

Indicate,  and  find  quotients  by  cancellation. 

2.  Divide  36  x  27  x  49  x  38  x  50  by  70  x  18  x  15. 

3.  (28  X  38  X  48)  -  (14  X  19  X  24  X  2  X  2)  =  ? 

4.  (26  X  5  X  54)  -  (13  X  5  X  6)  =  ? 

5.  What  is  the  quotient  of  36  x  48  x  16  divided  by 
27  X  24  X  8  ? 

6.  Divide  5  x  45  x  7  x  20  by  49  x  5  x  4  x  9. 

7.  Divide  5  x  51  x  7  x  9  x  4  by  17  x  20  x  12  x  7  x  2. 

8.  Divide  25  x  2  x  72  x  14  by  6  x  9  x  120. 

9.  How  many  bushels  of  potatoes  at  50  cents  a  bushel 
must  be  given  in  exchange  for  15  pounds  of  tea  at  40  cents 
a  pound  ? 

10.  If  60  yards  of  cloth  cost  $120,  how  many  yards 
can  be  bought  for  $40? 

11.  15  oranges  cost  45  cents.  How  much  will  7  oranges 
cost? 

12.  A  dairyman  sells  100  quarts  of  milk  daily  at  5  cents 
a  quart.  How  many  bushels  of  corn  at  45  cents  a  bushel  can 
he  buy  with  10  days'  milk  receipts  ? 

13.  A  farmer  sold  a  grocer  45  bushels  of  apples  at  50 
cents  a  bushel,  taking  his  pay  in  flour  at  90  cents  a  sack. 
How  many  sacks  did  he  receive  ? 

14.  In  what  time  can  a  boy,  at  60  cents  a  day,  earn  as 
much  as  a  man  can  earn  in  40  days  at  $  3  a  day. 

Note.  —  Change  the  $  3  to  cents. 

15.  If  32  quarts  of  chestnuts  cost  $  2.50,  what  will  800 
quarts  cost  ? 


WRITTEN   EXERCISES.  65 

16.  There  are  16  ounces  in  a  pound.  30  pounds  of  steel 
will  make  how  many  horseshoes,  each  weighing  6  ounces  ? 

17.  A  man  sold  15  cords  of  wood  at  $  6  a  cord  and 
received  payment  in  wheat  at  90  cents  a  bushel.  How 
many  bushels  of  wheat  did  he  receive? 

18.  If  I  buy  10  yards  of  cloth  at  f  2  a  yard,  and  pay  for 
it  in  wool  at  50  cents  a  pound,  how  many  pounds  of  wool 
will  it  require  ? 

19.  If  wood  is  worth  ^4  a  cord,  and  coal  $5  a  ton,  how 
many  cords  of  wood  will  pay  for  20  tons  of  coal  ? 

20.  Divide  the  product  of  18,  6,  9,  and  4  by  the  product 
of  10,  7,  6,  and  2. 

21.  Divide  the  product  of  10,  75,  9,  and  96  by  the  product 
of  5,  12,  15,  and  9. 

22.  Find  the  quotient  of  51  times  the  product  of  54  and 
12  divided  by  36  times  the  product  of  17  and  3. 

23.  How  many  jars  of  butter  each  containing  10  pounds 
at  20  cents  a  pound  must  be  given  for  10  sacks  of  granulated 
sugar  each  containing  5  pounds  at  5  cents  a  pound  ? 

24.  A  grocer  sold  20  boxes  of  soap,  each  containing  100 
packages  at  4  cents  a  package,  and  took  in  payment  hay  at 
^  16  a  ton.     How  many  tons  did  he  receive  ? 

GREATEST  COMMON  DIVISOR. 

103.  1.  What  number  will  exactly  divide  12  and  15? 
12  and  36  ?     15  and  20  ? 

Note.  —  When  two  or  more  numbers  have  the  same  factor,  it  is 
called  a  common  factor  of  those  numbers. 

2.  Name  a  common  factor  of  9,  12,  and  15. 

3.  What  factor  is  common  to  15,  20,  and  25  ?  What 
factor  is  common  to  14,  21,  and  28? 

4.  Name  two  common  divisors  of  10  and  20. 

5.  Name  the  greatest  factor  that  is  common  to  both  18 
and  30. 


66  GREATEST    COMMON   DIVISOll. 

104.  A  number  that  is  a  factor  of  two  or  more  numbers 
is  called  a  Common  Divisor  of  them. 

Thus,  5  is  a  common  divisor  of  10  and  15. 

105.  The  greatest  factor  of  each  of  two  or  more  numbers 
is  called  the  Greatest  Common  Divisor  of  them. 

Thus,  6  is  the  greatest  common  divisor  of  18  and  24. 

106.  When  two  or  more  numbers  have  no  common  factor 
or  divisor,  they  are  Prime  to  each  other. 

Thus,  8  and  15  are  prime  to  each  other. 

107.  Principle.  —  The  greatest  common  divisor  of  two 
or  more  numbers  is  the  product  of  all  their  common  prime 
factors.  V,. 

Written. 

1.    What  is  the  greatest  common  divisor  of  90  and  150. 
90  =  3x3x5x2 


Solution.  —  The  prime  factors  common 
to  both  60  and  150  are  2,  3,  and  5.  And 
since  the  greatest  common  divisor  of  two 
or  more  numbers  is  tlie  product  of  their 
common  prime  factors,  30  is  the  greatest 
common  divisor  of  90  and  150. 


150  =  2x5x5x3 

2  X  3  X  5  =  30,  Ans. 

OR 

2 

90     150 

5 

45      75 

3 

9       15 

3        5 

2  X  3  > 

:  5  =  30,  Ans. 

T'ind  the  greatest  common  divisor : 

2.  84,  132  7.    40,    60,    80  12.  45,    60,    90 

3.  63,    42  8.    64,  144,  560  13.  36,    72,    81 

4.  90,  105  9.    36,    48,    24  14.  44,  121,  132 

5.  112,  168  10.   40,    56,    72  15.  63,  126,  189 

6.  132,  156  11.    18,    54,    32  16.  36,    81,  135 


WRITTEN   EXERCISES.  67 

108.    To  find  the  greatest  common  divisor  when  the  numbers 
cannot  be  readily  factored. 

17.    What  is  the  greatest  common  divisor  of  510  and  935  ? 

Solution.  —  The  greatest  common  divisor  must  be  a  factor  of  both 

these  numbers.     It  cannot  be  the  larger.     It  is  not  the  smaller,  for 

we  find   a  remainder  of  425 
610)935(1  after  dividing   the  larger  by 

gj^Q  the  smaller. 

If  the  remainder  425  is  a 

425)510(1  factor  of  510,  it  will  be  the 

425  greatest    common    divisor   of 

_      ^    ^  -^  ,..  .     ~77\.or/i?      425  and  510,  and  therefore  of 

Greatest  Common  Divisor  85)425(5      ^-.^        -,  nor      tj  ^   v   • 

/        ^        610  and  935.     But  it  is  not, 

^^'^  for  we  find  a  remainder  of  85 

after  dividing  610  by  426. 

If  the  remainder  85  is  a  factor  of  425,  it  will  be  the  greatest  common 

divisor  of  itself  and  425,  also  of  426  and  610  ;  also  of  510  and  986.    We 

find  that  85  is  a  divisor  of  425.     It  is  therefore  the  greatest  common 

divisor  of  510  and  935. 

Note  1.  — An  exact  divisor  of  a  number  is  an  exact  divisor  of  any 
number  of  times  that  number. 

Note  2.  — An  exact  divisor  of  each  of  two  numbers  is  an  exact  divisor 
of  their  sum  and  of  their  difference. 

Rule.  —  Divide  the  greater  number  by  the  smaller,  and  the  last 
divisor  by  the  last  remainder  until  there  is  no  remainder. 
Tlie  last  divisor  will  be  the  greatest  common  divisor. 
If  more  than  two  numbers  are  given,  find  the  greatest  common 
divisor  of  two  of  them,  then  of  this  divisor  and  a  third 
number,  and  so  on.  The  last  divisor  will  be  the  greatest 
common  divisor. 
Find  the  greatest  common  divisor : 


18. 

270,810-" 

22. 

504,  560  f  ^      26.    120,  180,  240 

19. 

360,  420 

23. 

646,950  ^^       27.    140,280,420 

20. 

294,  567  . ' 

24. 

216,  324  1^  <     28.   288,432,1152 

21. 

264,  312j4 

25. 

300,  480  fpO      29.    225,  360,  405 

30. 

Find  the  greatest 

common  divisor  of  72,  153,  315, 

2187. 

(\ 

68  LEAST   COMMON  MULTIPLE. 

31.  I  have  32  bushels  of  wheat,  48  of  barley,  and  128 
of  oats.  I  desire  to  put  all  this  grain  into  boxes  of  the 
largest  possible  size,  so  that  no  box  shall  contain  more  than 
one  kind  of  grain.  How  many  bushels  must  each  box  con- 
tain? \i. 

There  will  be  how  many  boxes  of  wheat?  Of  barley? 
Of  oats  ? 

LEAST  COMMON  MULTIPLE. 

109.  1.  Name  a  number  of  which  3  is  a  factor.  Of  which 
5  is  a  factor. 

2.  Name  several  numbers  that  are  exactly  divisible  by  2. 
By  7.     By  5. 

3.  Name  a  number  that  is  exactly  divisible  by  both  3  and 
2.  Name  another.  Another.  Another.  What  is  the  least 
number  that  is  exactly  divisible  by  3  and  2  ? 

4.  What  is  the  smallest  number  that  will  exactly  contain 
5  and  6  ? 

110.  A  Multiple  of  a  number  is  a  number  that  exactly 
contains  it. 

Thus,  5,  10,  and  15  are  multiples  of  5. 

Note.  —  Pupils  sometimes  mistake  multiples  for  factors. 

A  multiple  is  a  product.     A  factor  is  a  divisor. 

111.  A  Common  Multiple  of  two  or  more  numbers  is  any 
number  that  exactly  contains  each  of  them. 

Thus,  60  is  a  common  multiple  of  4,  5,  and  6. 

112.  The  Least  Common  Multiple  of  two  or  more  numbers 
is  the  smallest  number  that  exactly  contains  each  of  them. 

Thus,  30  is  the  least  common  multiple  of  3,  5,  and  6. 


DBB^INITIONS.  69 

113.  Principle.  — The  least  common  multiple  of  two  or 
more  numbers  is  the  product  of  all  the  prime  factors  *in  the 
largest  number  multiplied  by  the  product  of  such  prime 
factors  of  the  other  numbers  as  are  not  found  in  the 
largest. 

114.  Written. 

1.    What  is  the  least  common  multiple  of  21,  28,  and  30  ? 

Solution.  —  Separating  the  numbers  into  their  prime  factors,  and 
multiplying  the  product  of  the  prime  factors  of  the  largest  number, 

2  X  3  X  5  =  30,  by  the  product  of  the 
21  =  3  X  7  prime  factors  of  the  other  numbers  not 

Qo  _  2  y  2  y  7  found  in  the  largest,  we  have  2x7  =  14. 

QA  ~  o       Q       K  Therefore  14  x  30  =  420,  the  least  com- 

30  =  2  X  3  X  O  jjjQjj  multiple.     The  prime  factors  that 

2x3x5x2x7=  420.      enter  into  this  least  common  multiple 

are  2,  3,  5,  2,  7. 
The  least  common  multiple  must  contain  all  the  factors  of  30 
(2  X  3  X  5)  or  it  would  not  contain  30.  It  must  contain  the  prime 
factors  of  21  (3  x  7).  3  is  also  a  prime  factor  of  30,  and  is  not  again 
included,  but  7,  not  being  a  factor  of  30,  must  be  included  in  the  least 
common  multiple,  or  it  will  not  contain  21. 

Of  the  prime  factors  of  28  (2x2  x7),  there  are  two  2's.  Since 
the  largest  number  has  but  one  factor  2,  the  factor  2  must  be  again 
included  in  the  least  common  multiple,  or  it  would  not  contain  28. 
The  factor  7  is  excl\ided  because  it  is  also  a  factor  of  21.  We  now 
find  that  all  the  factors  of  the  three  numbers  are  found  among  the 
factors  of  the  least  common  multiple  2x3x5x2x7. 

The  practical  method  of  finding  the  least  common  multiple 
is  as  follows : 

Solution. — We  divide^ by  any  prime  number  that  is  contained 
in  two  or  more  of  them,  and  the  quo- 
tients and  undivided  numbers  again 
in  like  manner,  until  the  remaining 
quotients  are  prime  to  each  other. 
"J       2       5  The  product  of  all  the  divisors  and 

the  last    quotients  will    be  the  least 
2x3x7x2x5  =  420.     common  multiple. 


2 

21 

28 

30 

3 

21 

14 

15 

7 

7 

14 

5 

70  LEAST   COMMON   MULTIPLE. 

Find  the  least  common  multiple : 

2.  18,  27,  30        6.  36,  40,  48  10.  24,  42,  54,  360 

3.  9,  12,  18  7.  18,  24,  36  11.  25,  20,  35,  40 

4.  16,  48,  60        8.  15,  30,  21,  28  12.  14,  21,  35,  45 
6.  21,  27,  36        9.  15,  60,  140,  210  13.  24,  48,  96,  192 

14.  Find  the  contents  of  the  smallest  box  that  may  be 
filled  with  wheat  by  using  a  4-quart,  a  5-quart,  or  a  6-quart 
measure.  How  many  4-quart  measures  will  fill  it  ?  How 
many  5-quart  measures  ?     6-quart  ? 

15.  Three  boys  ride  around  a  circular  track.  A  goes 
around  once  in  5  minutes,  B  once  in  8  minutes,  C  once  in 
10  minutes.  If  they  start  together,  how  many  minutes 
must  elapse  before  they  all  come  together  at  the  starting- 
point  ?  How  many  times  will  each  have  gone  around  the 
circle  ? 

115.  Review  of  Factors,  Multiples,  Divisors,  and  Cancel- 
-iation. 

1.  Define  factor,  composite  number,  prime  number,  and 
prime  factor. 

2.  Find  the  prime  factors  of  5075;  of  9576;  of  3150;  of 
6006. 

3.'  Find  the  sum  of  the  prime  factors  of  34650. 

4.  Find  the  prime  factors  of  2310  ;  of  17199 ;  of  6840. 

5.  81158  is  the  product  of  what  prime  factors  ? 

6.  Find  the  largest  prime  factor  of  12600. 

7.  What  is  a  common  divisor  of  two  or  more  numbers  ? 

8.  What  is  the  greatest  common  divisor  of  two  or  more 
numbers  ? 

9.  When  are  numbers  prime  to  each  other  ? 


REVIEW.  71 

Find  the  greatest  common  divisor  of : 

10.  672  and  960.  13.    1650  and  1920. 

11.  616  and  1012.  14.    696,  1218,  and  1160. 

12.  272  and  428.  15.   450,  720,  and  810. 

16.  What  is  the  greatest  prime  factor  common  to  4242 
and  2626  ? 

17.  A  grocer  had  84  bananas  and  126  lemons,  which  he 
wished  to  put  into  bags,  each  bag  containing  the  largest 
number  possible,  and  each  containing  the  same  number. 
How  many  could  be  put  into  each  bag  ? 

18.  A  man  Has  three  fields  containing  respectively  14, 18, 
and  22  acres.  He  wishes  to  cut  them  into  the  largest  pos- 
sible lots  of  equal  size.  How  much  land  will  each  lot  con- 
tain ?     How  many  lots  will  each  field  contain  ? 

19.  What  is  a  multiple  of  a  number  ?  A  common  mul- 
tiple ?     The  least  common  multiple  ? 

Find  the  least  common  multiple  of : 

20.  96,  196,  42,  and  54.  23.    252,  462,  and  1092. 

21.  45,  36,  70,  and  90.  24.    120,  280,  and  308. 

22.  36,  40,  42,  and  48.  25.    36,  110,  98,  and  66. 

26.  Find  the  least  common  multiple  of  the  even  numbers 
to  and  including  20. 

27.  What  is  the  least  sum  with  which  I  can  buy  an  exact 
number  of  chairs  at  $  6,  $  8,  or  $  5  each  ? 

28.  What  is  the  smallest  sum  of  money  that  may  be 
expended  by  using  an  exact  number  of  nickels,  dimes,  quar- 
ters, or  3-cent  pieces  ? 

How  many  pieces  of  each  kind  will  the  sum  contain  ? 

29.  John  can  run  around  a  block  in  6  minutes,  James  in 
8  minutes,  and  Henry  in  9  minutes.  If  they  start  together, 
how  long  before  they  will  all  be  together  again  at  the 
starting-point  ? 


28 

x56 

x30 

14x3 

x5 

28 

x32 

x7 

u 

x35 

x2 

34 

X    9 

x5 

72  EEVIEW   OF  FACTORS. 

30.  What  is  the  shortest  piece  of  rope  that  can  be  cut 
into  pieces  32,  36,  and  44  feet  long  ? 

31.  What  is  cancellation  ? 

32.  Of  what  use  is  cancellation  ? 

Find  results  of  the  following  by  cancellation : 

33.  ^^  X  '^6  X  48 

24  X  6  X  12 

34.  lO^xAxi  37 

4x6 

35.  ^^X^X^^X^  '  38. 

6x8x2  25x17x3 

39.  240  X  48  X  70  X  18  --42  x  15  x  54  X  7  =  ? 

40.  Divide  the  product  of  25,  14,  and  11  by  the  product 
of  15,  7,  and  22. 

41.  How  many  bushels  of  wheat  at  $1.10  a  bushel  must 
be  given  for  6  pieces  of  cloth  each  containing  33  yards  at 
50  cents  a  yard  ? 

42.  How  many  cords  of  wood  at  $3  a  cord  will  pay  for 
30  lb.  of  sugar  at  5  cents  a  pound  ? 

43.  If  8  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  12  men  do  it  ? 

44.  How  many  pounds  of  sugar  can  be  bought  for  $7  if 
21  lb.  cost  $1.05? 

45.  How  many  pounds  of  maple  sugar  at  12  cents  a 
pound  must  a  farmer  exchange  for  15  pounds  of  coffee  at 
24  cents  a  pound  ? 

46.  A  milkman  exchanges  8  cans  of  milk,  30  quarts  in  a 
can,  at  4  cents  a  quart,  for  3  pieces  of  sheeting,  40  yards  in 
a  piece.     What  is  the  price  of  the  sheeting  per  yard  ? 


COMMON   FRACTIONS. 


116.  1.  When  any  whole  thing,  as  an  apple,  is  divided 
into  two  equal  pieces,  what  part  of  the  whole  will  each 
piece  be  ? 

2.  If  anything  is  divided  into  3  equal  parts,  what  is 
each  part  called  ?  Into  4  equal  parts  ?  Into  5  equal  parts  ? 
Into  8  equal  parts  ? 

3.  One  of  the  two  equal  parts  of  an  apple  is  one  half  of 
it.     One  half  is  written  ^. 

4.  One  of  the  two  equal  parts  of  a  number  is  \  of  the 
number. 

5.  How  many  are  ^  ^f  4  oranges  ?  -I-  of  20  cents  ?  -J 
of  10  ? 

6.  One  of  the  three  equal  parts  of  anything  is  one-third 
of  it.  One-third  is  written  \.  How  many  are  ^  of  6  men  ? 
\  of  12  dollars  ?     i  of  18  days  ?     i  of  24  ? 

117.  One  or  more  of  the  equal  parts  of  a  unit  is  called  a 
Fraction. 

The  unit  of  which  the  fraction  is  a  part  is  called  the 
Unit  of  the  Fraction. 

One  of  the  equal  parts  is  called  a  Fractional  Unit. 

Two  or  more  fractions  having  the  same  fractional  unit 
are  Like  Fractions. 

73 


I  f 


74  COMMON   FRACTIONS. 

118.  A  Fraction  is  written  with  two  numbers,  one  above 
the  other,  with  a  line  between  them ;  as  \. 

The  number  below  the  line  is  the  Denominator,  and  it 
shows  into  how  many  equal  parts  the  unit  is  divided. 

Thus,  in  the  fraction  -|,  8  is  the  denominator. 

The  number  above  the  line  is  the  Numerator,  and  it  shows 
how  many  of  the  parts  are  taken. 

'-    Thus,  in  the  fraction  f ,  5  is  the  numerator.         xy^^tJ^^n^ 
^  r    '^  ^  V   '^f  li6  numerator  and  denominator  are  called  the  Terms  of 
a  Fraction. 

Thus,  3  and  4  are  the  terms  of  f. 

^  119.  A  Common  Fraction  is  a  fraction  written  with  its 
numerator  above  its  denominator  with*  a  line  between. 

•'  120.  A  Proper  Fraction  is  a  fraction  whose  value  is  less 
than  1.      Its  numerator  is  less  than  its  denominator;   as 

4     3      7 

7?  ¥>   8- 

*  121.  An  Improper  Fraction  is  a  fraction  whose  value  is  1. 
or  more  than  1.  Its  numerator  is  equal  to  or  greater  than 
its  denominator ;  as  |,  |,  -y-. 

122.  An  integer  may  be  written  in  fractional  form  by 
giving  it  1  for  a  denominator,  when  it  becomes  an  improper 
fraction. 

Thus,  5  =  1,  4  =  f 

•^  123.   A  number  composed  of  an  integer  and  a  fraction  is 
a  Mixed  Number. 

Thus,  31   12|. 

124.    A  fraction  is  indicated  division,  the  numerator  being 
the  dividend,  and  the  denominator  the  divisor. 
Thus,  I  means  3  h-  4.     -i^  means  12  -r-  3. 


REDUCTION   OF    FKACTIONS.  _  75 

125.    The   Value   of   a   Fraction   is   the    quotient   of   the 
numerator  divided  by  the  denominator. 

Write  in  figures : 

7.  Four  sevenths.  10.  Seventeen  eighteenths. 

8.  Five  eighths.  11.  One  twenty -fourth. 

9.  Nine  sixteenths.  12.  Eight  forty-seconds. 

13.  Nine  and  seven  tenths. 

14.  Twenty-five  and  eight  elevenths. 

15.  Fourteen  and  seven  ninths. 


Eead  the  following : 


16. 

\ 

19. 

\\ 

22. 

¥ 

25. 

f 

28. 

17. 

fV 

20. 

A 

23. 

\\ 

26. 

If 

29. 

18. 

3%V 

21. 

1 

24. 

1  6 
2'3 

27. 

1  5 
9""0 

30. 

120 
TTTOT 


3 


_2_ 

1"(5"1 


REDUCTION   OF  FRACTIONS. 

1.  In  one   apple   there   are   how  many   halves  ?      How 
many  fourths  ?     How  many  eighths  f 

2.  In  |-  of  an  apple,  how  many  fourths?      How   many 
eighths  f 

3.  How  many  eighths  in  J  of  an  apple  ? 

4.  In  one  apple  there  are  how  many  thirds  f    How  many 
sixths  ? 

5.  In  1^  there  are  how  many  sixths  f    How  many  imiths  9 
In  J  how  many  sixths  f     How  many  ninths  f 

6.  Name  a  fraction  that  is  equal  to  i.     Name  a  fraction 
equal  to  \.     To  \. 

7.  Change  |  to  halves.     Change  |  to  thirds. 

8.  Change  |^  to  fourths.     To  eighths. 


76 


COMMON  FRACTIONS. 


9.    Change  ^  to  sixths.     To  ninths. 

10.  Change  |  to  sixths.     To  ninths. 

11.  Express  f  in  larger  terms.     What  operations  did  you 
perform  ? 

Express  |  in  smaller  terms.     What  did  you  do  ? 
Reduction  of  fractions  is  the  process  of  changing  their 
form  without  changing  their  value. 

126.   To  higher  terms. 

12.  Eeduce  f  to  sixteenths. 

Solution.  —  Since  we  must  change  4ths  to  16ths,  the  new  denomi- 
nator must  be  4  times  the  given  denominator.     Since 

3x4 12     tlie  new  denominator  will   be   4   times  the    given 

4x4      16     denominator,  the  new  numerator  must  be  4  times 
the  given  numerator.     Therefore,  multiplying  both 
terms  of  the  fraction  |  by  4  gives  i|. 


J     Principle.  —  Multiplying  both  terms  of  a  fraction  by  the 
same  number  does  not  change  the  value  of  the  fraction. 


Change  the  following : 

13.  ftolOths  19.    fto27ths 

14.  I  to  9ths 

15.  ito30ths 
16. 


f  to  18ths 


17.    |tol2ths 


18. 


I  to  24ths 


20. 


I  to  56ths 


21.  -i-to48ths 

22.  fto21sts 

23.  ^  to  50ths 

24.  i5_to22ds 


f  to  84ths 


25.  4  to  25ths 
26. 

27.  ii  to  eOths 

28.  I  to  72ds 

29.  1  to  63ds 

30.  |to40ths 


To  reduce  a  fraction  to  higher  terms. 

Rule. — Multiply   both   terms   of  the   fraction  by  the   same 
*        number. 


Note.  — To  find  the  multiplier,  divide  the  required  denominator  by 
the  given  denominator. 


REDUCTION    OF    FRACTIONS. 


77 


Written. 

Change  the  following: 

31.  iJto96ths       35.  ||tol40ths  39.  |f  to  500ths 

32.  I  to  64ths         36.  ^  to  150ths  40.  |-J  to  168ths 

33.  ^%  to  75ths       37.  If  to  144ths  41.  ij  to  522ds 

34.  i|tol20ths     38.  fi-tol28ths  42.  iff  to  9375ths 


127.    To  lowest  terms. 

A  fraction  is  expressed  in  its  lowest  terms  when  the  terms 
are  prime  to  each  other. 

1.    Change  ||  to  lowest  terms. 

Solution.  —  Since  we  must  change  16ths  to  4ths,  the  new  denomi- 
nator must  be  i  of  the  given  denominator.     Since  the 
1_  -j-  4  _  3     new  denominator  will  be  I  of  the  given  denominator, 
16-5-4      4     the  new  numerator  must  be  I  the  given  numerator. 
Therefore,  dividing  both  terms  of  ^f  by  4  gives  |. 

^      Principle.  —  Dividing  both  terms  of  a  fraction  by  the 
same  number  does  not  change  the  value  of  the  fraction. 

Change  to  lowest  terms  : 


2. 

f 

6. 

3. 

H 

7. 

4. 

1 

8. 

5. 

A 

9. 

11 

1  8 


56 
6T 


42 


10. 

11. 

12. 
13. 


.3  6 

T2 


24 


H 


14. 
15. 
16. 
17. 


1  5 
2T 


18 
■2¥ 


45 
¥¥ 

if 


18. 

u 

19. 

u 

20. 

M 

21. 

u 

y 


Written. 
^^^Jl^^  Reduce  |^  to  lowest  terms. 


—  "^     =-  Solution. — Dividing  both  terms  of  |-f  by  5  we 

4o  -5-  o      y  have  f .     Dividing  both  terms  of  f  by  3  we  have  |. 

6-5-3      2  Since  the  terms  of  f  are  prime  to  each  other,  they 

Q  _s_  3      3  are  the  lowest  terms  of  f^.     .  a  /.  ^   ./  .  ^  . 


78  COMMON    FRACTIONS. 

To  reduce  a  fraction  to  its  lowest  terms. 
Rule.  —  Divide  both  terms  by  any  common  factor,  and  divide 

the  result  in  the  same  way  until  the  terms  are  prime  to 

each  other. 
If  the  terms  are  large  numbers,  divide  by  their  greatest 

common  divisor. 

Eeduce  to  lowest  terms : 


23. 

II 

30. 

t\\ 

37. 

tW^ 

44. 

t¥A 

24. 

tVs 

31. 

iH 

38. 

-Hi 

45. 

m 

25. 

a 

32. 

2% 

39. 

m 

46. 

AV 

26. 

n 

33. 

m 

40. 

tW 

47. 

t¥t 

27. 

If 

34. 

m 

41. 

»¥t 

48. 

iff 

28. 

5  5 

35. 

2¥^ 

42. 

m 

49. 

iff 

29. 

aV^ 

36. 

m 

43. 

m\ 

50. 

m 

51.  How  many  thirds  in  ffl? 

52.  Express  in  simplest  form  98  divided  by  392. 

53.  Change  -^y-^j  to  a  fraction  whose  denominator  is  5. 

54.  Express  in  simplest  form  the  quotient  of  288  divided 
by  504. 

55.  What  is  the  simplest  form  of  -J-||  ? 

128.    Integers  and  mixed  numbers  to  improper  fractions. 

1.  In  2  apples  how  many  halves?     How  many  fourths? 
How  msLTij  fifths? 

2.  In  5  apples  how  many  halves?     How  many  fourths? 
fifths?  eighths? 

3.  How  many  4ths  in  2  ?  in  8  ?  in  10  ?  in  15  ?  in  20  ? 

4.  How  many  fifths  in  1  ?  in  2  ?  in  2i  ?  in  3|  ?  in  4f  ? 

5.  Keduce  5 J  to  halves.     7^  to  eighths.     4|-  to  sixths. 
4^  to  sevenths. 

6.  Change  4i  to  9ths.     3|  to  3ds.     5^%  to  lOths.     8f  to 
5ths.     7t\  to  llths. 


REDUCTION    OF   FRACTIONS.  79 

Reduce  to  improper  fractions  : 


7. 

^ 

14. 

n 

21. 

8A 

28. 

m 

8. 

H 

15. 

H 

22. 

H 

29. 

m 

9. 

3f 

16. 

H 

23. 

lOf 

30. 

111 

10. 

H 

17. 

n 

24. 

iH 

31. 

n 

11. 

H 

18. 

H 

25. 

12f 

32. 

H 

12. 

n 

19. 

H 

26. 

lOA 

33. 

12| 

13. 

4f 

20. 

^A 

27. 

ii« 

34. 

10| 

35.  How  many  9ths  in  13  ? 

36.  Change  26  to  a  fractio^ial  form. 

$7.  How  many  fifths  in*  32  ?  in  24  ?  in  16  ?  in  50  ? 

38.  In  5^  weeks  how  many  7ths  of  a  week  ? 

39.  What  improper  fraction  is  equal  to  11|^? 

Written. 

40.  Reduce  25  to  fifths. 

^^    .  ^        J  Solution.  —  In  1  there  are  ^.     In  25  there 

Zo  times  ^  =  -^-     jj^^g^  ^^  25  times  f ,  or  i|^. 

41.  Change  28  to  sevenths. 

42.  Reduce  16^  to  an  improper  fraction. 

16^ 

7  sevenths  Solution.  —  Since  in  1  there  are  ^,  in 

112  16  there  are  16  times  ^j  or  ip. 

4  sevenths 


116  sevenths,  =  i^ 


6 


To   reduce   an   integer   or  mixed   number  to   an    improper 
fraction. 

/  Rule.  —  Multiply  the  integer  by  the  denominator,  add  the  nu- 
merator of  the  fraction,  if  any,  and  write  the  residt  over 
the  denominator. 


80         •  COMMON   FRACTIONS. 

Reduce  to  improper  fractions : 


43. 

25i 

50. 

35tV 

57. 

59f 

64. 

238| 

44. 

i^fi 

51. 

m 

58. 

67f 

65. 

359A 

45. 

35A 

52. 

251 

59. 

4|? 

66. 

iio^VV 

46. 

49tV 

53. 

I^tV 

60. 

5if 

67. 

483^^ 

47. 

270f 

54. 

27A 

61. 

29A 

68. 

846ff 

48. 

19J 

55. 

4H 

62. 

i^H 

69. 

359ff 

49. 

28A 

56. 

253V 

63. 

l^A 

70. 

482if 

71.  In  560  how  many  otlis  ? 

72.  Reduce  250  to  IGtlis.     349  to  15ths. 

73.  Change  12|  to  16ths.     ^4|  to  18ths. 

74.  In  $  730  how  many  fourths  of  a  dollar  ? 

75.  Change  156  to  a  fraction  whose  denominator  shall 
be  12. 

129.    Improper  fractions  to  integers  or  mixed  numbers. 

1 .  How  many  dollars  in  4  quarter-dollars  ?    In  8  quarter- 
dollars  ?     In  16  quarter-dollars  ?     In  20  quarter-dollars  ? 

2.  How  many  dollars  in  $  -2/  ?     In  ^  Y_  ?     In  f  2_8  9 

3.  12  fourths  of  a  bushel  are  equal  to  how  many  bushels  ? 
36  fourths  ?     40  fourths  ? 

4.  4  fourths  of  a  dollar  are  equal  to  how  many  dollars  ? 

5.  To  how  many  dollars  are  8  fourths  of  a  dollar  equal? 
9  fourths  ?     11  fourths  ? 

Reduce  to  integers  or  mixed  numbers : 


6. 

f 

11. 

i 

16. 

¥ 

21. 

¥ 

26. 

H 

7. 

1 

12. 

¥ 

17. 

¥ 

22. 

124 

27. 

ff 

8. 

f 

13. 

¥ 

18. 

¥ 

23. 

a 

28. 

M 

9. 

i 

14. 

¥ 

19. 

80 
1  1 

24. 

46. 
1  1 

29 

W 

0. 

-¥ 

15. 

¥ 

20. 

¥ 

25. 

a 

30.  In  -3^  of  a  pound  how  many  pounds  ? 

31.  In  -2^1^  of  a  dollar  how  many  dollars? 


/ 


REDUCTION   OF   FRACTIONS.  81 

Written. 

32.    Reduce  -^^^  to  an  integer  or  mixed  number. 

16  385'*'' 

■'ort  S0T.UT10N, — Since  ^f  equal  1,  W^-  will  equal    as 

— -—  many  times  1  as  16  is  c'bntained  in  385,  or  24^  times. 

65 
54  ^^i^s.  24tV. 

~1 
To  reduce  an  improper  fraction  to  an  integer  or  mixed  number. 
Rule.  —  Divide  the  numerator  by  the  denominator. 
Reduce  to  integers  or  mixed  numbers : 


33. 

¥ 

40. 

-w- 

47. 

Hi^ 

54. 

3M^63 

34. 

^      • 

>       41. 

w 

48. 

4973 

55. 

i_9_^69 

35. 

W 

42. 

w 

49. 

X2J4 

56. 

8_8_^2 

36. 

H 

43. 

-w- 

50. 

•  Hr- 

57. 

38_0_00 

37. 

If 

44. 

w 

51. 

3|.24 

58. 

25001, 

38. 

H 

45. 

\¥ 

52. 

1806 

59. 

-W/^ 

39. 

a 

46. 

-w 

53. 

3^|2 

60. 

8_y^6^ 

61.    How  many  bushels  in  ^^^  bushels  ? 

130.   Fractions  to  fractions  having  a  common  denominator. 

1.  How  many  eighths  in  1  ?     In  \  ?     In  J  ? 

2.  How  many  sixths  in  1  ?     In  ^  ?     In  J  ? 

3.  How  many  twelfths  in  i  ?     |  ?     i  ? 

4.  Write  1  and  ^  as  twelfths.     As  sixths. 

5.  Write  \  and  ^  as  eighths.     As  twelfths. 

6.  Change    \    and    ^    so    both    may  have   20    for    a 
denominator. 

7.  Change  -J-,  i,  and  ^  each  to  12ths. 

8.  Change  |  and  J  each  to  24ths. 

v  When  fractions  have  the  same  denominator  they  are 
Like  Fractions,  and  their  denominator  is  called  a  Common 
Denominator. 

Thus,  ^J,  ^J,  and  |^}  have  a  common  denominator. 


82  COMMON   FRACTIONS. 

131.  The  smallest  common  denominator  of  two  or  more 
fractions  is  their  Least  Common  Denominator. 

Thus,  If,  ^|,  and  if  become  y^^,  y%,  and  ■^-^,  when  changed 
to  their  least  common  denominator. 

The  common  denominator  of  two  or  more  fractions  is  a 
common  multiple  "of  their  denominators. 

The  least  common  denominator  of  two  or  more  fractions 
is  the  least  common  multiple  of  their  denominators. 

Written. 
1.    Reduce  J  and  |-  to  fractions  having  a  common  denomi- 
nator. 

3x6 18  Solution. — The  common   denominator   must  be   a 

4  x^  g      24     common  multiple  of  the  denominators  4   and  6,  and 
since  24  is  the  product  of  the  denominators  it  is  a  com- 
_  '*^  '*  _.  zH     mon   multiple  of  them.     Therefore  24   is   a  common 
6x4      24     denominator  of  |  and  |. 

f=|f,andf  =  ff. 
Beduce  to  fractions  having  a  common  denominator : 

q5  11  ifil234 


2. 

hi 

3. 

hi 

4. 

hi 

5. 

hi 

6. 

hi 

7. 

hi 

8. 

hi 

1       1       1 

3"?        4?       Z 


3_      5 

12?     9' 


10.  I,  f,     f  17. 

11.  h  h   I-  18- 

19  5  2  1  1  q  4         i        _8_  _7_ 

13.  j\,  1      i  20.  i,     i,     ^^,  -/_ 

14.  h  if  21.  y%,  f,     I,  I 

1  K  7  5  4  22  i         _2_     _3_  i 

■'■°-  ¥?  6?        9"  '^'''  2?        11?     13'  3 

23.    Eeduce  |,  f,  and  -^-^  to  fractions  having  the  least 
common  denominator. 


^?  ^>  1^  Solution.  —  The    least    common    denominator 

3,  3,  6  must  be  the  least  common  multiple  of  the  denomi- 


1,  1,  2  nators  3,  6,  and  12,  which  is  12. 

2x3x2  =  12  j  =  ^         l  =  IS         ^,  =  A 


30. 

h 

h  h 

1 

31. 

h 

tV  rf  tt? 

f 

32. 

h 

IZ'     2"T' 

H 

REDUCTION   OF   FRACTIONS.  83 

To  reduce  fractions  to  fractions  having  a  common  denominator. 

Rule.  —  Reduce    each  fraction    to   its    lowest    terms,    divide 

the  least  common  midtiple  of  the   denoininators  by  the 

denominator  of  each  fraction,  and  midtiply  both  terms  by 

the  quotient. 

Note.  — If  the  denominators  are  prime  to  each  other,  their  product 

will  be  their  common  denominator. 

Reduce  to  fractions  having  the  least  common  denominator : 

25.  I,      h      fV 

26.  h    h    tV 

97        4  5  3 

28.    T%   2h  f  33.    f  ^\,  A,     4 

34.    Which  fraction  is  larger,  ^  or  J  ? 


132.    Oral. 

1.  Add  land  i      |.,  |,  and  f      |,  |,  and  f .      |-,  f,  and  f . 

2.  rind  the  sum  of  f,  J,  and  f     Qf  f,  ^,  and  f .     Of  |,  |, 
and  |. 

3.  Find  the  sum  of  f ,  |,  |,  and  -i-.     Of  fV?  tV  A?  and  ^^g-. 
^f  tVj  tVj  tV.  and  |i. 

4.  In  ^  how  many  fourths  ?     Add  ^  and  i. 

5.  ^  equals   how   many   tenths?      ^   equal   how  many 
tenths  ?     Add  |  and  f . 

6.  ^  equals  how  many  sixths  ?     |-  ?     Add  ^  and  ^. 

7.  Find  the  sum  of  ^  and  \.     Of  i  and  ^.     Of  |  and  f 

8.  Add  i  I,  and  i.     Add  i  -|,  and  i.     Add  |,  |,  and  f 

9.  If  I  pay  J  of  a  dollar  for  breakfast  and  |-  of  a  dollar 
for  dinner,  what  will  both  meals  cost  me  ? 


84  COMMON  FRACTIONS. 

10.  A  boy  paid  f  of  a  dollar  for  a  book  and  :^  of  a  dollar 
for  paper.     How  much  did  he  pay  ? 

11.  A  owns  ^  of  a  store,  and  B  f .  How  many  eighths  do 
both  own  ? 

12.  John  saves  i  a  dollar  a  week,  and  Charles  f  of  a  dol- 
lar.    How  many  fourths  do  both  save  ? 

13.  Henry  gave  -\  of  his  marbles  to  one  boy  and  J  of 
them  to  another.     How  many  twelfths  do  both  receive  ? 

14.  A  clerk  sold  i  a  pound  of  tea  to  one  customer,  J  to 
another,  and  |-  to  another.     How  many  eighths  did  he  sell  ? 

15.  A  man  pays  ^  of  his  salary  for  rent,  ^  for  table 
expenses,  and  -f^  for  clothing.  What  part  of  his  money  was 
expended  for  rent,  table,  and  clothing  ? 

16.  A  farmer  planted  |  of  his  seed  in  one  field  and  f  of  it 
in  another.  What  fraction  represents  the  seed  planted  in 
both  fields  ? 

17.  A  man  paid  $2^  for  a  hat  and  f  4i  for  shoes. 
What  is  the  cost  of  both  ? 

18.  Jack  deposits  $  3\  in  the  bank,  Elsie  $  li,  and  Susie 
$  |.     How  much  do  all  deposit  ? 

Principle.  —  Only  like  fractions  can  be  added. 

Written. 

19.  What  is  the  sum  of  f,  |,  and  -^^  ? 

Solution.  —  The  least  common  multiple  of  the  denominators  is  48. 
l)ividing  this  by  the  denominator  of  each  fraction  and  multiplying 

both  terms  of  the  quotient, 

f +  f  +  T'6=if +  M  +  H  =  -W-      we    have    ff,    If,    and    ||. 

i_03  _  2  7.     Ans.  '^^^^  fractions  are   now  like 

fractions,  and  are  added  by 
adding  their  numerators  and  placing  the  sum  over  the  common  de- 
nominator.    Hence  the  sum  is  -L"^/,  or  2^^^. 


ADDITIOK.  85 

20.    What  is  the  sum  of  of,  Ty^^,  6^^  ? 

^f    ~  •-*  3Tr  Solution.  —  Since  there  are  both  integers  and  f rac- 

T-j^  =  7|^  tions  to  be  added,  we  find  first  the  sum  of  the  fractions, 

6rj^  =  6^-^  which  is  fl,  or  Iff.     This  is  added  to  the  sum  of  the 

192  3  integers,  18.     18  +  Iff  =  19|f .     Ans. 

3  0 

Rule.  — If  the  fractions  are  not  like  fractions,  reduce  them  to 
a  common  denominator,  add  their  numerators,  ayid  place 
the  sum  over  the  common  denominator.  Reduce  the  result 
to  lowest  terms.  If  the  result  Is  an  improper  fraction, 
reduce  it  to  an  integer  or  mixed  number. 
When  there  are  integers,  or  mixed  numbers,  add  the  frac- 
tions and  integers  separately,  and  unite  their  results. 

Find  the  sum  of : 

32.  1,1,8,1 

33.  i  3f,f,6 

34.  i,hn>T% 

35.  4|-,  y,  3,  -^ 

36.  |,i,TV6 

37.  7f,8f,f  I 

OQ       4.     2      5     1 

39.  7|,9i,6i,4| 

40.  lOi,  74,11,11 

41.  ^,^,l0i,48 

42.   19f,  18f,  15i,  12tV 

43.  A  man  walked  21J  miles  on  Monday,  27|  miles  on 
Tuesday,  and  28^*^  miles  on  Wednesday.  How  far  did  he 
walk  in  the  three  days  ? 

44.  A  farmer  owned  three  fields,  containing  respectively 
42^^  acres,  36f  acres,  and  322^^^  acres.  How  many  acres 
were  there  in  all  ? 


21. 

h  A,  i 

22. 

h  h  ^T 

23. 

h  h  A 

24. 

hhl 

25. 

A,  h  f 

26. 

iii.A 

27. 

h  A»  TdJ  i 

28. 

'S'?  ■§■?   12  >  g"!" 

29. 

h  h  h  i\ 

30. 

1%  h  h  T5 

31. 

¥?  T2"?   2"??  T¥ 

86  COMMON   FRACTIONS. 

46.  How  many  yards  of  cloth  in  four  pieces  containing 
13|-  yards,  21i  yards,  31|  yards,  and  45|  yards  ? 

46.  John  lives  241  rods  from  school,  Harry  6/g-  rods 
farther  than  John,  and  Thomas  lOjf  rods  farther  than 
Harry.     How  far  does  Thomas  live  from  school  ? 

47.  A  farmer  sold  hay  for  $  45J,  oats  for  $  15y%,  corn  for 
f  20|,  and  potatoes  for  $  35|.  What  was  the  amount  of 
his  sales  ? 

48.  I  paid  $  2^  for  car-fare,  $  4^  for  cotton  cloth,  $  5f  for 
shoes,  $  25^  for  a  suit  of  clothes,  and  had  $  1  ^^^  left.  How 
much  money  did  I  have  at  first  ? 

49.  A  boy  was  absent  from  school  the  first  week  of  the 
term  12i  hours,  the  second  week  16f  hours,  the  third  week 
8j7^  hours,  and  the  fourth  week  7^%  hours.  How  many  hours 
was  he  absent  during  the  four  weeks  ? 

50.  I  bought  31  tons  of  coal  in  January  for  $  19^,  2J  tons 
in  February  for  $  15|,  and  5j\  tons  in  March  for  $  30  j?^. 
How  much  coal  did  I  buy  in  all,  and  what  was  the  cost  ? 

61.  James  weighs  5S^^  pounds,  William  65|  pounds, 
Charles  67-J  pounds,  and  their  father  as  much  as  all  three 
of  them.     How  much  does  their  father  weigh  ? 

62.  I  bought  of  A  102%  *^^s  of  hay,  of  B  18|-%  tons,  and 
of  C  16|  tons.     How  many  tons  did  I  buy  in  all  ? 

63.  Four  piles  of  wood  contain  respectively  24|  cords, 
18f  cords,  27|-  cords,  and  30i-  cords.  How  many  cords  in 
all? 

SUBTRACTION. 


Oral. 

133.    1.    From  f 

take 

i- 

How  much 

is  1  less  f  ? 

i  less 

5 

?     ii-less^^? 

2.    From  |  take 

i- 

How  much  is  ^ 

minus  f  ?  | 

minus 

i 

?|-iV? 

SUBTRACTION.  -.  87 

3.  A  boy  had  $  ^^  and  spent  $  ^.    What  part  of  a  dollar 
had  he  left  ? 

4.  A  owns  I  of  a  store,  and  B  |.    How  much  of  the  store 
does  A  own  more  than  B  ? 

5.  John  runs  ^  of  a  mile,  and  Jerry  |  of  a  mile.    Which 
runs  farther,  and  what  part  of  a  mile  ? 

6.  Mr.  Ames  owned  |  of  a  farm,  and  sold  i  of  it.    What 
part  remained  ? 

7.  What  is  the  difference  between  |  of  anything  and  4 
of  it  ?     Which  is  greater  ? 

8.  Lucy  has  $  ^,  and  Alice  $  |.     Which  has  the  more, 
and  how  much  ? 

Subtraction. 

10.  f  -  i  15.     f  -  J  20.    4  -  I 

11.  i-i  16.    A-i  21.    2-l| 

12.  i-i  17.     J  -i  22.    5-3| 

13.  i-i  18.    J^-i  23.    4i-2i 
Principle.  —  Only  like  fractions  can  be  subtracted. 

Written. 

24.  From  f  subtract  |. 

Solution.  —  We  find  the  least  com- 
^  ~"  "g  —  T8  "■  Tt  —  TS'       mon  denominator  of  |  and  |  to  be  18. 
_3    ^  1       jij^s  6  =  t!'  and  I  =  tI-      Their  difference 

'  ^       '  is  r\  or  1. 

25.  From  11^  subtract  4|. 

Solution.  —  We  subtract  the  fractions  and  inte- 

111  __  10 s       gers  separately.     After  changing  the  fractions   to 

45  _    45       their  least  common  denominator,  we  have  11§  —  4|. 

— ~      I  cannot  be  subtracted  from   |,   hence  we  take  1 

6^      of  the  11  units,  change  it  to  sixths,  and  add  the  f , 

making  10|.  lOf  -  4|  =  6|  =  6^.     Ans. 


88  COMMON   FRACTIONS. 

Rnle.  —  If  the  fractions  are  not  like  fractions,  reduce  them  to 
a  common  denominator,  and  write  the  difference  of  their 
numerators  over  the  common  denominator. 
When  there  are  integers,  or  mixed  numbers^  subtract  the 
fractions  and  integers  separately. 

Note.  —  Mixed  numbers  may  be  changed  to  improper  fractions, 
and  subtracted  as  fractions. 


Subtraction. 

26.    i-l 

34. 

18i-4i 

42. 

h-i. 

27.    l-i 

35. 

16 -If 

43. 

ii-U 

28.    T^^-i 

36. 

16^-23 

44. 

M-H 

29.    A-f 

37. 

«-| 

45. 

M-A 

30.    16 -J 

38. 

M-l 

46. 

H-A 

31.    31 -i 

39. 

tV-I 

47. 

M-i 

32.    3i-i 

40. 

If-oV 

48. 

ii-^% 

33.    8-21 

41. 

\i-^\ 

49. 

M-M 

50.    From  3846J  take  2944f. 

Subtract : 

51.    58627 

52. 

3169 

f           53. 

98701 

54.    1000^2,. 

im^\ 

30502^ 

4963| 

358A 

55.  From  a  cask  of  oil  containing  42  gallons  I  sell  8i 
gallons,  17f  gallons,  and  5|  gallons.  How  much  oil  remains 
in  the  cask  ? 

56.  A  rod  is  16J  feet.     Take  12J  feet  from  a  rod. 

57.  From  a  cask  of  vinegar  containing  43|  gallons  17| 
gallons  were  drawn.     How  many  gallons  remained  ? 

58.  A  farmer  having  217  bushels  of  wheat,  sold  95JJ 
bushels.     How  many  bushels  had  he  left  ? 

69.  The  minuend  is  123 J^ J,  and  the  remainder  381^. 
What  is  the  subtrahend? 


MULTIPLICATION.  89 

60.  From  a  piece  of  cloth  containing  54|  yards,  15^^ 
yards  were  sold  at  one  time  and  21J  yards  at  another. 
How  many  yards  were  left? 

61.  If  a  grocer  gained  $  1|  by  selling  a  barrel  of  flour  for 
$  6  j,  what  did  it  cost  him  ? 

62.  A  lady  bought  a  hat  for  $  4|,  shoes  for  f  5|,  and 
some  cotton  cloth  for  $  3^-^,  and  gave  in  payment  a  twenty- 
dollar  bill.     How  much  change  should  she  receive  ? 


M  UL  TI  PLICA  TION. 
OraL 

134.    1.    4  times  1  apple  are  how  many  apples  ?     7  times 
1  apple  ?     4  times  6  apples  ? 

2.  5  times  1-ninth  are  how  many  ninths?     6  times  one- 
ninth  ?     3  times  |  ? 

3.  4  times  J  equals  how  many  6ths  ?     5  times  |?     8 
times  f  ? 

4.  5  times  y\  =  how  many  sixteenths.     5  times  ^j  —  how 
many  elevenths  ? 

5.  6  times  2%  =  ?     8  times  f  ?     10  times  | ? 

6.  3xJ=?     8xf  =  ?     4xi  =  ?     3xf  =  ? 

7.  How  much  is  ^  of  4  ?     i  of  6  ?     ^  of  10  ?     i-  of  12  ? 

8.  How  much  is  i  of  1  ?     ^ofi?     lofi? 

9.  How  much  is  i  of  1  ?     ^  of  ^  ?,    ^  of  1.     i  of  ^  ? 

10.  How  much  is  |  of  |^  of  an  orange  ?     ^^  of  |-  of  an 
orange? 

11.  At  $  f  each  what  will  8  books  cost  ? 

12.  If  a  horse  eat  |^  of  a  bushel  of  grain  in  a  week,  how 
much  will  5  horses  eat  ? 


90  COMMON   FRACTIONS. 

13.  If  a  pound  of  tea  costs  48  cents,  what  will  J  of  a 
pound  cost  ?     What  will  |  of  a  pound  cost  ? 

14.  A  owned  -|  of  a  ship,  and  sold  ^  of  his  share.  What 
part  of  the  ship  did  he  sell  ?     What  part  was  left  ? 

15.  What  will  i  yard  of  ribbon  cost  at  16  cents  a  yard? 
^  of  a  yard  ?     f  of  a  yard  ? 

16.  8  boys  earn  $  f  each.     What  do  all  earn  ? 

17.  John  earns  $f,  Henry  ^  as  much,  and  Edward  ^  as 
much.     What  do  Henry  and  Edward  earn  ? 

18.  John  earns  $  J,  and  Henry  |  as  much.  How  much 
does  Henry  earn  ? 

I  of  I  =  ?     Since  -^  of  f  =  i   |  will  be  2  times  i  or  |  =  ^. 

A71S. 
Written. 

19.  Multixjly  f  by  f. 

Solution.  —  i  multiplied  by  |  is  the  same  as  f 

§  X  -  =  —  =  -;     of  |.     ^  of  f  =  ^,  and  I  is  3  times  ^,  or  f.     This 

o      4      20       5      solution,  in  effect,  is  the  same  as  multiplying  the 

J      3      3      numerators  together  for  a  new  numerator,   and 

^^f      K  ^  J  ~  K*     the  denominators  for  a  new  denominator. 

Cancellation  shortens  the  process. 

-»/  Rule.  —  Reduce   integers   and  mixed  numbers  to   improper 

fractions. 
Multiply  the  numerators  together  for  the  numerator  of  the 

product,  and  the  denominators  for  the  denominator  of 

the  product. 
Cancel  when  possible. 

Two  or  more  fractions  joined  by  of  form  a  Compound 
Fraction.  The  word  of  between  two  fractions  is  equivalent 
to  the  sign  of  multiplication. 

To  change  a  compound  to  a  simple  fraction,  multiply  the 
fractions  together. 
Thus,!  of  |of  |  =  |x|x|  =  |. 


MULTIPLICATION.  9l 

Find  the  products : 


20. 

|x  J 

29. 

i  Of  i  of  1 

38. 

5|  X  24  X  20 

21. 

|x| 

30. 

f  of  ,%  of  t 

39. 

7.}  X  5f  X  f 

22. 

Axf 

31. 

¥  X  A  X  f 

40. 

9i  X  3V  X  2i 

23. 

J  of  A 

32. 

fV  X  1  X  1 

41. 

H  X  ii  X  A 

24. 

t  of  U 

33. 

Ax^xf 

42. 

3V  X  4  X  51. 

25. 

tV  X  T  4 

34. 

\^  X  34  X  1 

43. 

i\  X  80  X  5i 

26. 

if  xf 

35. 

It  X  90  X  i 

44. 

3^  X  51  X  T^ 

27. 

ifxA 

36. 

8xf  xf 

45. 

f  X  16  X  If 

28. 

iVx^^ 

37. 

16  X  1  X  1 

46. 

45  X  41  X  t\ 

Find  the  value : 

47. 

1  of  40 

51. 

1  of  328 

55. 

ij  of  342 

48. 

f  of  42 

52. 

f  of  721 

56. 

If  of  800 

49. 

1  of  16 

53. 

4  of  90 

57. 

j\  of  2222 

50. 

A  of  17 

54. 

f  of  131 

58. 

1  of  1632 

59.  Find  the  product  of  124|  by  5. 

124f 

5 
Solution. — Multiplying  the  fraction  and  inte- 

•^t  ger  separately  by  5,  we  have  5  times  |=  J^  =  3f, 

^^Q  and  5  times  124  =  620.     620  +  3|  =  623|.  Ans. 
623f 

5  X  f  =  -V-  =  3f 

Find  the  products : 

60.  13i  X  4  65.  16f  X  30      70.  95|  x  45 

61.  184  X  10  66.  21-1  X  29      71.  64|  x  81 

62.  16f  X  5  67.  451J  X  15      72.  84f  X  16 

63.  28f  X  9  68.  48J  X  63      73.  34/y  x  63 

64.  48f  x^8  69.  24|  X  25      74.  4}  x  18 


92  COMMOJT  FRACTIONS. 

75.    Find  the  product  of  127  x  4J. 

127 
43 

— ^  Solution.  —  Multiplying  by  the  fraction  and 

integer  separately,  we  have  f  of  127  =  95^.   4  times 
127  =  608.     508  +  95^  =  GOSJ.    Ans. 


95} 
508 

603} 

of  127  =  95} 

Multiply : 

76.    65  by  7| 

77.    45  by  34 

78.    83  by  5f 

79.    72  X  16| 

80.    84xlli^ 

81.  26  by  9f  86.  3156  by  ^ 

82.  64  by  5J  87.  8165  by  7f 

83.  89  by  5|  88.  4950  by  9| 

84.  56  by  4f  89.  2835  by  16| 

85.  92  by  lOf  90.  5872  by  25f 

91.  ^offoff  of  |oft  =  ?  ^.^^of^of-Vof  A  =  ? 

92.  T^off|of,-«^ofifof|  =  ?  |ofifof|of||  =  ? 

93.  II  X  J  of  -f  of  ii  of  42  =  ?  51  X  f  of  ^\  of  I  =  ? 

94.  4f  of  if  of  fl  of  if  of  I  of  if  of  5j\  =  ? 

95.  Mr.  Brown  earns  $40 J  a  month,  and  his  son  |  as 
much.     How  much  does  the  son  earn  ? 

96.  At  $  12|  a  ton,  how  much  will  9^-^  tons  of  hay  cost? 

97.  What  will  be  the  cost  of  48f  yards  of  cloth  at  $  f  a 
yard  ? 

98.  A  man  gave  124Y^g-  acres  of  land  to  his  two  sons,  giv- 
ing f  of  it  to  the  elder  and  f  to  the  younger.  How  many 
acres  did  eaxih  receive  ? 

99.  If  it  requires  21|  days  for  a  man  to  dig  a  ditch,  in 
what  time  can  he  dig  ^  of  it  ? 

100.  A  man  owning  Jg-  of  a  cotton  mill  sold  -fl-  of  his 
share.     What  part  of  the  mill  did  he  sell  ? 


MULTIPLICATION.    '  93 

101.  In  a  school  containing  945  pupils,  j-  of  the  number 
were  boys.     How  many  boys  in  the  school  ? 

102.  What  is  the  cost  of  15^  acres  of  land  at  $45|  an 
acre? 

103.  Mr.  Clark,  owning  f  of  a  farm  of  128  acres,  sold  his 
share  at  $  45^  an  acre.  How  much  should  he  receive  from 
the  sale  ? 

104.  At  ^  J  a  bushel,  what  will  |  of  f  of  a  bushel  of 
wheat  cost  ? 

105.  A  lady  purchased  lOf  yards  of  silk  at  $1|-  a  yard. 
What  was  the  cost  ? 

106.  I  paid  $150  for  a  horse,  and  ^J-  as  much  for  a 
carriage.     What  did  the  carriage  cost  ? 

107.  What  will  be  the  cost  of  a  side  of  beef,  containing 
252  pounds,  at  9^  cents  a  pound  ? 

108.  The  divisor  is  15f  and  the  quotient  21|.  What  is 
the  dividend  ? 

DIVISION, 

135.    1.    How  many  times  is  J  contained  in  1  ? 

Solution.  —  1  equals  f .  Therefore  ^  is  contained  in  f ,  4  times. 
t-^i  =  4. 

2.  How  many  times  is  J  contained  in  1  ? 

Solution.  — Since  ^  is  contained  in  1,  4  times,  f  is  contained  in  It 
I  as  many  times.     ^  of  4  times  is  2  times,     f  -^  f  =  2. 

3.  How  many  times  is  f  contained  in  1  ? 

Solution.  — Since  \  is  contained  in  1,  4  times,  |  is  contained  in  It 
I  as  many  times.     ^  of  4  times  equals  f  times. 

4.  How  many  times  is  ^  contained  in  1  ?    |  in  1  ?    ^  in 
1?     finl?     finl? 


94  COMMON    FRACTIONS. 

5.  ^  js  contained  in  2  how  many  times  ? 
Solution. — 2  equals  |.     | -r- 1- =  6. 

6.  I  is  contained  in  2  how  many  times  ? 

7.  2-r-i  =  ?     4--i  =  ?     3--|  =  ?     3--f  =  ? 

8.  If  you  earn  f  of  a  dollar  in  a  day,  how  long  will  it 
take  to  earn  7  dollars  ? 

9.  At  ^  I  apiece  how  many  books  can  be  bought  for  $  6  ? 

10.  If  I  save  $  f  a  day,  in  how  many  days  can  I  save 

$8? 

11.  How  much  is  4  divided  by  2? 

Solution.  —  f  -^  2  is  the  same  as  ^  of  |.     ^  of  |  =  |.     Therefore 

fH-2  =  |. 

12.  If  3  balls  are  worth  -^^  of  a  dollar,  what  will  1  ball 
be  worth  ? 

13.  What  will  one  book  cost  if  3  books  cost  |  of  a 
dollar  ? 

14.  I  divided  -^j  of  my  money  equally  among  4  boys. 
What  part  did  each  boy  receive  ? 

15.  Divide  f  by  3.     i^  by  5.     if  by  5. 

16.  I  is  contained  in  |  how  many  times  ? 

Solution.  —  f  is  contained  in  1,  f  times.  Therefore  it  is  contained 
in  f ,  I  of  f  times,  or  f  times. 

Written. 

17.  I  is  contained  in  f  how  many  times  ? 

Solution.  — |  is  contained  in  1,  |  times.  Since  f  is  contained  | 
times  in  1,  in  I  it  is  contained  f  of  f  times,  or  |  times.  Thus,  we  see 
that  the  divisor  |  has  become  inverted,  and  multipUcation  performed. 

^  Rule.  —  Multiply  the  dividend  by  the  divisor  inverted. 
Cancel  when  possible. 

Note.  —  Change  integers  and  mixed  numbers  to  improper  fractions. 


DIVISION. 


95 


Find  the  quotients : 

18.  JH-I       23.    S^^i-i 

19.  H^l       24.    6i^/^ 


Oft         5     _s_   3 


25.    -^^5f 


y^^^ 


21.  if -I       26.    10 

22.  H^ft     27.    11^51 
38.    4  X  3^5  of  3  =  ? 


28. 

29.  8 

30.  10 

31.  i 


32.    ii 


5. 

6 

14 

8 


33.  2|-5-5^ 

34.  7i-^li 

35.  23H-H 

36.  2J--3i 

37.  8|H-9| 


Solution.  —  Inverting  the  divisor,  indicating  the  operations,  and 
cancelling,  we  have 

-•       A71S. 

5     ^     ^     3     5 


4      3      0 
-  X  -  X  ^  X  - 


39.  f  of  9 -J- f  of  6- 

40.  4ofi|^|of4 

41.  i7^x|^Aof22 


42. 


TT 


-tVx 


X* 


43.  3^-1  X  J  of  2 

44.  4x5x3^7| 

45.  Hx||xH-H 

46.  8ix5i^|of| 


47.    Divide  31563  by  5. 


5)31561(6312^ 
30 

~T5  lf  =  J 

6 
5 

If 


Solution.  —  When  the  integer  of  a  mixed 
number  is  large,  it  may  be  divided  as  folio v^^s : 
5  is  contained  in  3156|,  631  times,  with  a 
remainder  of  1|.  This  remainder  being 
divided  by  5,  gives  -^^,  which  we  place  at 
the  right  of  the  quotient. 


Find  the  quotients : 

48.  47f--7 

49.  384f-!-5 

50.  287^9j^~8 
61.  3854-5-5 


52.  139871 -I- 9 

53.  897243  H- 15 

54.  69834^-- 24 

55.  969851^25 


96  COMMON  FRACTIONS. 


56.    Divide  3682  by  51 

Solution.  —  When  the  dividend  contains  several  figures  and  the 
Ki \ Qgco  divisor  is  a  mixed  number  it  is  often  more  convenient 
2  2        ^^  divide  as  above. 

TT  YT^OA  ^^  multiply  both  dividend  and  divisor  by  2,  when 

i the  divisor  becomes  11  (halves),  and  the  dividend  7364 

boy  j-y    (halves).     Dividing,  the  quotient  is  669 j\. 

Principle.  —  Multiplying  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 

3f ind  the  quotients : 

Si 


57.  356 --4^  61.   39846 

58.  728 --8i  62.    44077 


71- 


59.  397 --5J  63.    76582 

60.  296-- 101-  64.    28769 

65.    If  16  bushels  of  apples  cost  $  8|,  vrhat  will  1  bushel 
cost? 


66.  Five  heirs  shared  equally  in  the  division  of  a  legacy 
of  $  35,862|.     What  was  the  share  of  each  ? 

67.  When  15  bushels  of  wheat  sell  for  $  17f,  what  is  the 
price  per  bushel  ? 

68.  The  product  of  two  numbers  is  326|.     One  of  the 
numbers  is  5.     What  is  the  other  ? 

69.  There  are  5^  yards  in  a  rod.     How  many  rods  in 
3158  yards  ? 

70.  If  a  man  walks  15|^  miles  a  day,  in  how  many  days 
can  he  walk  155  miles  ? 

71.  What  is  the  price  of  coal  per  ton  when  16  tons  cost 
$73f? 

72.  How  much  does  a  man  earn  in  a  day  if  he  earns 
$  84J  in  a  month  of  26  days  ? 


DIVISION.  97 

73.  When  flour  is  $  6|  per  barrel,  how  many  barrels  can 
be  bought  for  $  297  ? 

74.  At  6^  cents  apiece  how  many  tablets  can  be  bought 
for  $  5  ? 

75.  If  coffee  is  37^  cents  a  pound,  how  many  pounds  can 
be  bought  for  f  60  ? 

76.  If  a  boy  can  read  17J  pages  of  a  book  in  an  hour,  in 
how  many  hours  can  he  read  175  pages  ? 

77.  If  IJ  bushels  of  corn  cost  $1.^*^,  what  will  1  bushel 
cost? 

78.  How  many  books  can  be  bought  for  $31^,  if  1  book 
[costs  $  3^  ? 

>J    A  fraction  having  a  fraction  for  one  or  both  of  its  terms 
is  called  a  Complex  Fraction. 

Note.  —  Like  all  fractions,  it  is  an  expression  of  division. 

42 

79.  Eeduce  -^  to  a  simple  fraction. 

75  ^ 

4|     -M 
Solution.  —  7!  =  A-     Since  all  fractions  indicate  the  division  of  the 

numerator  by  the  denominator,  JI  means  1^  -^  ^. 

Dividing,  we  have  ^-^  x  5^7  =  ff .     Ans. 

Therefore  to  simplify  a  complex  fraction,  divide  the  tiumerator  by 
the  denominator. 

Change  to  simple  fractions : 

80.  li         82.    13        84.    ii        86.    5i         88.    i^LI 
«  4  16f  J^  I  of  I 

81.  m      83.    «  85.    i  87.    ^         89.    i^L3 

i    ,  16  A  A  J 

90.   If  f  of  an  acre  of  land  is  worth  $  72^,  what  is  the 
value  of  an  acre  at  the  same  rate  ? 


9B  COMMON   FRACTIONS. 

91.  There  are  5 J  yards  in  a  rod.  How  many  rods  in  70|- 
yards  ? 

92.  At  ^  5  J  a  ton,  how  many  tons  of  coal  can  be  bought 
for  I  73^  ? 

93.  If  I  of  a  yard  of  silk  costs  $  |,  what  will  1  yard  cost? 

94.  How  many  bags  will  be  needed  to  hold  92^-  bushels 
of  wheat,  if  1  bag  holds  2i  bushels  ? 

95.  At  $  f  per  bushel,  how  many  bushels  of  corn  can  be 
bought  for  $  62^  ? 

96.  The  product  of  three  fractions  is  ^^g,  and  two  of  them 
are  f  and  ii.     What  is  the  third. 

97.  What  will  12|-  yards  of  broadcloth  cost  if  |  of  a  yard 
costs  $  41  ? 

THE    THREE    QUESTIONS   OF  RELATION, 

136.  1.    3  times  4  equals  what  ?     A71S.  12. 

2.  12  is  how  many  times  4  ?     Ans.  3. 

3.  12  is  3  times  what  ?     An,s.  4. 

In  question  1,  we  have  two  factors,  to  find  their  product. 
In  questions  2  and  3,  we  have  the  product  and  one  factor, 
to  find  the  other. 

1.  Form  questions  like  2  and  3,  from  the  following  state- 
ment: 5  X  6  =  30. 

a.    |-  of  8  equals  what  ? 

Multiplying  8  by  |-,  we  have  4.     Ans. 

h.   4  is  ^  of  what  ? 

Since  ^x8  =  4,  4-5-^  =  8.     Ans. 

c.   4  is  what  part  of  8  ? 

Since  |x8  =  4,  4-8  =  ^.     Ans. 


QUESTIONS   OF   RELATION.  99 

Principle.  —  The  product  of  two  numbers  divided  by  one 
of  them  gives  the  other. 

To  THE  Teacher.  —  In  such  examples  as  question  a,  after  the 
product  is  found,  it  may  be  used  with  each  of  the  two  numbers  to  form, 
successively,  question  b  and  question  c.  Drill  upon  these  three  ques- 
tions of  relation  should  be  so  thorough  that  each  question  will  suggest 
its  own  solution  instantly. 

2.  i  of  24  =  what  ?     (Question  a.) 

3.  After  finding  the  product  in  example  2,  form  question 
b.     Question  c. 

4.  8  is  1^  of  what  ?     (Question  b.) 

Solution.  —  From  the  question  it  is  evident  that  8  is  the  product  of 
two  numbers,  and  that  ^  is  one  of  them.  Therefore,  8  -^  ^  =  24.  8  is 
I  of  24. 

5.  What  part  of  24  is  8  ?     (Question  c.) 

Solution.  —  It  is  evident  that  8  is  the  product  of  two  numbers,  and 
24  is  one  of  them.     Therefore,  8  -4-  24  =  -^j  or  f     8  is  |  of  24. 

Question  a. 
Find  result,  and  form  questions  b  and  c: 

6.  How  much  is  f  of  12  ?  9.    f  of  15  =  ? 

7.  How  much  is  f  of  16  ?  10.    ^  of  21  =  ? 

8.  How  much  is  I  of  20  ?  11.    |  of  40  =  ? 

Question  b. 
Find  result,  and  form  questions  a  and  c : 

12.  15  is  f  of  what?  15.    18  is -f- of  what  ? 

13.  4  is  I  of  what  ?  16.    24  is  |  of  what  ? 

14.  9  is  f  of  what  ?  17.    25  is  |  of  what  ? 


100  COMMON   FRACTIONS. 

Question  c. 
Find  result,  and  form  questions  a  and  h  \ 

18.  What  part  of  24  is  8  ?    21.    21  is  what  part  of  35  ? 

19.  What  part  of  18  is  12  ?    22.    28  is  what  part  of  63  ? 

20.  What  part  of  9  is  2  ?      23.    15  is  what  part  of  25  ? 

Find  result,  form  the  other  two  questions,  and  solve  each : 

24.  I  of  bQ  equals  what?      28.    How  much  is  -^^  of  96  ? 

25.  What  part  of  49  is  14  ?    29.    38  is  y2_  of  what  number  ? 

26.  26  is  I  of  what  ?  30.    16  is  what  part  of  80  ? 

27.  64  is  what  part  of  120?    31.    18  is  y%  of  what  number  ? 

Remark.  —  Each  of  the  following  problems  contains  one  or  more 
of  the  three  questions  of  relation.  Before  attempting  to  solve  any  of 
them,  the  pupil  should  state  the  question  in  each  of  them. 

32.  James  had  56  marbles,  and  John  |  as  many.  How 
many  had  John  ? 

The  question  is.  How  much  is  f  of  56  ?  —  a. 

33.  John  had  42  marbles,  which  was  f  as  many  as  James 
had.     How  many  had  James  ? 

The  question  is,  42  is  j  of  what  ?  —  h. 

34.  James  had  56  marbles,  and  John  42.  John's  marbles 
are  what  part  of  James's  ? 

The  question  is.  What  part  of  56  is  42  ?  —  c. 

35.  A  man  sold  50  acres  of  land,  which  was  ^  of  all  he 
had.     How  many  acres  had  he  at  first  ? 

36.  A  boy  had  20  cents  *and  spent  15  cents.  What  part 
of  his  money  did  he  spend  ?     What  part  was  left  ? 

37.  Mr.  A  has  640  sheep,  and  Mr.  B  -^^  as  many.  How 
many  has  Mr.  B  ? 

38.  f  of  a  ton  of  hay  cost  $  12.  What  was  the  cost  of  a 
ton? 


QUESTIONS    OF   RELATION.  101 

39.  I  of  a  basket  of  eggs  were  sold  for  f  6.  What  was 
the  value  of  the  entire  basket  ? 

40.  At  $  45  an  acre,  how  much  land  will  $  25  buy  ? 

41.  If  I  of  a  factory  is  worth  $  6300,  what  is  the  value  of 
the  factory  ? 

42.  A  man.  who  owed  $  7825  failed,  and  could  pay  only  | 
of  his  debts.     How  much  could  he  pay  ? 

43.  A  man  lost  ^  of  his  money  and  had  $  210  left.  How 
much  had  he  at  first  ? 

44.  If  -^-^  of  a  merchant's  capital  is  $  35,000,  what  is  his 
entire  capital  ? 

45.  If  a  man  can  do  a  piece  of  work  in  24  days,  what 
part  of  it  can  he  do  in  18  days  ? 

46.  In  my  pasture  are  75  sheep,  which  is  f  of  all  my 
sheep.     How  many  sheep  have  I  ? 

47.  Henry  runs  540  yards,  which  is  ^  as  far  as  Frank 
runs.     How  far  does  Frank  run  ? 

48.  A  barrel  can  be  filled  by  a  pipe  in  40  minutes. 
What  part  of  a  barrel  can  be  filled  in  25  minutes  ? 

49.  A  bushel  contains  32  quarts,  and  a  peck  8  quarts. 
What  part  of  a  bushel  is  a  peck  ? 

50.  I  bought  a  house  and  lot,  and  made  a  payment  of 
$  4500,  which  was  f  of  the  cost.  What  was  the  cost  of 
the  property  ? 

MISCELLANEOUS  REVIEW  OF  COMMON  FRACTIONS. 

137.   Oral. 

1.  Add  {  and  ^ ;  -^  and  ^ ;  |  and  | ;  -f  and  | ;  f  and  | ; 
i  and  If 


102  COMMON    FRACTIONS. 

3-  Reduce  to  improper  fractions  3^,  7|,  8|,  5|-,  7^,  8|, 
16|,  15i. 

4.  Reduce  to  integers  or  mixed  numbers  -\^-,  -y-,  J^,  -U-, 

¥,  ¥,  f  i  ¥tS  -VV- 

5.  Multiply  16  by  |;  45  by  f ;  18  by  |;  45  by  j\', 
I  by  9;    iby  32;    ||  by  16;    J  by  27. 

6.  Find  product  of:  f  x  ^;  A  X  y^^  5  ^J  X  ^5  f  X  if; 
^x-V-;4Jx6|. 

7.  Find  |  of  24;  f  of  12;  |  of  30;  f  of  27 ;  f  of  45; 
I  of  40. 

8.  Find  i  of  i;  i  of  |;  ^  of  f;  ^-^  of  f ;  |  of  If; 
i  of  21. 

9.  Divide  f  by  3;  f  by  4;  -^  by  12;  ^e,  by  11;  4^  by 
3;  ^by  6;  4|by6;  10  by  f ;  8  by  f ;  If  by  8;  Aby22; 
fby9. 

10.  Divide  4  by  i;  8  by  4;  9  by  f ;  16  by  f ;  24  by  |; 
13  by  I;  llbyt;  12  by  If. 

11.  Divide  1  by  J;   f  by  |;   i  by  i;   |  by  f ;  j\  by  |; 
fbyf;  51.  by  21 

12.  Divide  1  by  : 

111111  132 

3>    "§"?    6'    I'S"?    T>    Y:5">   TIT'    T'    "S"* 

Note.  —  1  divided  by  a  fraction  equals  that  fraction  inverted. 

13.  I  of  12=?     9  =  what  part  of  12 ?     9  is  J  of  what? 

14        Qy?_2.  A  -^9  —  L'  3y?_l.  5._:_9—  1- 

3_s_9_l.  5_:^9  —   8 

■3"    •    ^  —  "5"  5  ¥    •     •    —  "9  • 

15.  i-?  =  i;     i  +  ?  =  i;     !-?  =  !;     l  +  ?  =  ii; 

16.  What  part  of 

6  is  4  ?  i  is  i  ?  f  is  I  ?  5^  is  2^  ? 


1  ? 


11  is  5?  i  is  i?  lis  J?  |is     ^ 

17.    9  is  I  of  what  ?      5  is  |  of  what  ?      6  is  f  of  what  ? 


REVIEW   OF   ERACTIONS.  103 

18.  Change  f  to  24ths ;  |  to  loths;  |-  to  32ds;  f  to 
20ths. 

19.  A  man  owning  J  of  a  farm,  sold  ^  of  his  share. 
What  part  did  he  sell  ?     How  much  remains  ? 

20.  At  121^  a  dozen,  how  many  dozen  of  eggs  can  I 
buy  for  $3? 

21.  At  6J^  a  box,  how  much  will  8  boxes  of  berries  cost  ? 

22.  John  has  56  cents,  and  James  |-  as  much.  How 
much  have  both? 

23.  A  can  do  a  piece  of  work  in  4  days ;  B  can  do  the 
same  piece  of  work  in  2  days.  What  part  of  the  work  can 
each  do  in  a  day  ? 

24.  A  can  mow  a  field  in  3  days,  and  B  in  4  days.  What 
part  of  the  field  can  they  mow  in  a  day  if  both  work 
together  ? 

A  can  mow  \  of  it  in  I'day,  and  B  can  mow  ^  of  it  in  1  day. 
Both  working  together  can  mow  the  sum  of  ^  and  ^  =  -^^  of  it  in  1 
day. 

25.  C  can  do  a  piece  of  work  in  2  days,  and  D  can  do  it 
in  4  days.  In  what  time  can  they  both  do  it,  working 
together  ? 

C  does  I  of  it  in  1  day,  and  D  ^  of  it  in  1  day.  Therefore  both 
can  do  ^  +  ^  =  f  of  it  in  1  day.  Since  both  can  do  |  of  it  in  1  day, 
it  will  take  as  many  days  to  do  |,  or  the  whole  of  it,  as  |  is  contained 
times  in  |,  or  1^  days.     A7is. 

Note.  — |  divided  by  f  gives  the  same  result  as  4  divided  by  3. 

26.  ^  of  my  money  is  gold,  and  ^  as  much  is  silver. 
What  part  of  my  money  is  silver  ? 

27.  If  a  boy  can  earn  $  2i  in  1  week,  how  much  can  3 
boys  earn  in  4  weeks  ? 

28.  James  sold  a  book  for  28  cents,  which  was  |-  of  what 
it  cost  him.     What  did  it  cost  him  ? 


104  COMMON   FRACTIONS. 

29.  The  difference  between  ^  of  a  number  and  \  of  it  is  6. 
What  is  the  number  ? 

Solution.  —  The  difference  between  |  and  {  is  {.  Now  the  ques- 
tion is,  6  is  ^  of  what  ? 

30.  A  boy  12  years  of  age  is  \  as  old  as  his  father. 
How  old  is  his  father  ? 

Written. 

31.  A  farmer  having  1200  bushels  of  potatoes  sold  i  of 
them  at  one  time,  ^  at  another,  and  350  bushels  at  another. 
How  many  bushels  had  he  left  ? 

32.  A  mechanic  whose  wages  are  $5  per  day  uses  -^  of 
his  weekly  earnings  for  board,  and  |  for  clothing  and  other 
expenses.     How  many  dollars  does  he  save  weekly  ? 

33.  Which  is  greater  and  how  much,  yf  (3r  -^  ? 

34.  If  it  takes  27  days  to  do  a  piece  of  work,  how  long 
will  it  take  to  do  -|  of  it  ? 

35.  If  a  horse  is  worth  $100,  and  a  cow  is  worth  |-  as 
much  as  the  horse,  what  is  the  cow  worth  ? 

36.  John  has  in  the  bank  $45  and  draws  out  f  of  it. 
How  much  remains  in  the  bank  ? 

37.  What  will  16  pair  of  shoes  cost  at  $  3|^  a  pair  ? 

38.  If  a  farmer  has  23  sheep  and  sells  them  at  $3/^ 
apiece,  how  much  does  he  receive  for  the  sheep  ? 

39.  What  is  the  cost  of  -^-^  of  a  pound  of  cheese  at  10  ^  a 
pound  ? 

40.  What  is  the  cost  of  |  of  a  yard  of  cloth  at  $  IJ  a 
yard? 

41.  f  xf  x6ixy\x3xlf  =  ? 

42.  What  is  the  value  of  3 1  of  8|  of  |  of  J|  ? 

43.  A  man  sold  3f  tons  of  hay  at  one  time,  7|-  at  another, 
and  enough  the  third  time  to  make  20  tons.  How  many 
tons  did  he  sell  the  third  time  ? 


EEVIEW    OF   FKACTIONS.  105 

44.  jij  plus  ^  plus  J  plus  ^  and  how  many  more  will 
make  3  ? 

45.  A  man  having  a  farm  of  96  acres  sold  ^  of  an  acre  to 
one  man,  ^  of  an  acre  to  another,  ^  of  an  acre  to  another, 
and  -^j  of  an  acre  to  another.     How  many  acres  had  he  left? 

46.  If  two  men  were  90  miles  apart,  and  each  should 
travel  23^  miles  toward  the  other,  how  many  miles  would 
they  then  be  apart  ? 

47.  If  George  has  ^  of  a  dollar  and  -^  of  a  dollar,  and 
Henry  has  J  of  a  dollar  and  ^^  of  a  dollar,  which  has  the 
greater  amount,  and  how  much  ? 

48.  A  man  bought  3  loads  of  wood  containing  respectively 
IJ  cords,  1|  cords,  and  1|  cords.  How  many  cords  of  wood 
did  he  buy  ? 

49.  I  paid  $101  for  hay,  f  lof  for  coal,  and  $6i  for 
wood.     What  did  I  pay  for  ail  ? 

50.  Mr.  Jones  paid  $  525J  for  a  span  of  horses,  and  sold 
them  for  $  6251-.     How  much  did  he  gain  ? 


51.  L.  W.  and  J.  E.  Connell  paid  f  4500f  for  a  store  and 
its  contents.  They  sold  it  for  $5025f.  How  much  did 
they  gain  by  the  operation  ? 

52.  A,  B,  C,  D,  and  E  own  respectively  ^,  |,  |,  -fi^,  and 
ii  acres  of  land.     How  much  do  they  all  own  ? 

53.  A  gentleman  having  f  1700  paid  $825J  for  horses, 
$230|  for  cows,  $1.50 J  for  oxen,  and  $407|-  for  sheep. 
How  much  money  had  he  left  ? 

54.  Mr.  Blanchard  paid  $  8^^  for  shovelling  his  walk, 
$  5|-  for  trimming  his  grape-vines,  and  $  6|  for  sifting  his 
ashes.  He  gave  the  man  a  20-dollar  bill  and  a  dollar  bill. 
How  much  money  should  Mr.  B.  receive  in  return? 

55.  If  I  add  2  to  each  term  of  the  fraction  ^,  will  its 
value  be  increased  or  diuimished;  and  how  much  ? 


106  COMMOl^   FRACTIONS. 

56.  Mr.  Homer  has  lOi  acres  of  wheat,  6|  acres  of  corn, 
20|  acres  of  barley,  and  16|  acres  of  rye.  How  many  acres 
of  grain  has  he  ? 

57.  What  is  the  quotient  of  389  divided  by  1556,  ex- 
pressed in  its  simplest  form  ? 


58. 


I  of  if  of  ^9_  off 


69.    816  is  f  of  what  number  ? 

60.  From  f  of  |f  take  f 

61.  The  product  of  two  factors  is  10^;  one  factor  is  3f. 
What  is  the  other  ? 

62.  3+624-9i  +  ^6_  +  _7_  =  9 

63.  (j\  of  21  of  j\)  X  (I  of  3i  of  8  X  J)  =  ? 

64.  The  sum  of  two  numbers  is  19||.  One  of  the  num- 
bers is  12f .     What  is  the  other  ? 

43 

65.  Reduce  -1  to  its  lowest  terms. 

28 

66.  a-i)xa+i)=? 

67.  Change  to  simple  fractions : 

if  H  tBii    n    fofi  If  L5LI. 

9'    5i'       16   '    fof2i'    J  of  J'    jj'    J  of  I 

68.  A  can  do  a  piece  of  work  in  4  days,  B  can  do  the 
same  work  in  5  days,  and  C  in  6  days.  In  what  time  can 
all  do  it  together  ? 

69.  A  tank  has  3  supply  pipes.  It  can  be  filled  in  6 
hours  by  the  first  pipe,  in  7  hours  by  the  second,  and  in 
8  hours  by  the  third.  In  how  many  hours  can  the  tank  be 
filled  by  the  three  pipes  together  ? 

70.  A  and  B  can  do  a  piece  of  work  in  3  days.  A  can 
do  it  alone  in  54  days.     In  what  time  can  B  do  it  alone  ? 


REVIEW    OF   FKACTIONS.  107 

Solution.  — Both  can  do  ^  of  it  in  1  day.  A,  alone,  can  do  j^  of 
it  in  1  day.  ^  —  j\  =  -^■^,  the  part  A  can  do  in  1  day.  Since  he  can 
do  ^5  of  it  in  a  day,  he  can  do  f|,  or  the  whole  of  it,  in  as  many  days 
as  j\  is  contained  times  in  f f,  or  33  -=-  5  =  6f  days. 

71.  J  of  my  property  is  invested  in  land,  f  of  the  re- 
mainder in  business,  and  |  of  the  remainder,  which  is 
$2400,  is  in  the  bank.     How  much  property  have  I? 

72.  What  is  the  value  of  (^1 J  +  6  -  f  of  f  +  i^  -  3i  ? 

73.  A  farmer  sold  11  doz.  eggs  at  14^^  a  dozen,  and  took 
his  pay  in  sugar  at  5|  ^  a  pound.    How  much  did  he  receive  ? 

74.  Find  the  value  of  — |—  -f-  i  of  i  --  ^. 

I  off       2        3       3 

75.  A  boy  having  spent  ^  of  |  of  his  money  for  a  knife, 
had  $  2.25  left.     How  much  did  he  pay  for  the  knife  ? 

76.  A  father  left  $  39,000  to  his  two  children,  dividing  it 
so  that  the  daughter  received  |  as  much  as  the  son.  What 
was  the  share  of  each  ? 

77.  A  person  owning  f  of  a  steamboat,  sold  f  of  his  share 
for  $  17360»     What  was  the  value  of  the  boat  ? 

78.  After  spending  i  of  my  money  and  J  of  the  remain- 
der I  had  $300  left.     How  much  had  I  at  first  ? 

79.  If  i  of  I  of  a  bushel  of  apples  cost  f  of  y^^  of  a  dol- 
lar, what  will  J  of  j  of  a  bushel  cost  ? 

80.  How  many  pounds  of  honey  at  |^  of  f  of  a  dollar  a 
pound  can  be  bought  for  |  of  2|  dollars  ? 

81.  Simplify  ri-y^-^^- 

12      •  2  X  3J      • 


DECIMAL   FRACTIONS. 


138.  A  Power  is  the  product  of  equal  factors,  as  5  x  5  =  25, 
5x5x5  =  125.  25  is  the  second  power  of  5.  125  is  the 
third  power  of  5.  10  x  10  =  100.  10  x  10  x  10  =  1000. 
100  is  the  second  power  of  10.     1000  is  the  third  power  of  10. 

>     139.    A  Decimal  Fraction  or  Decimal  is  a  fraction  whose 
denominator  is  10  or  a  power  of  10. 

y  Note.  —  The  denominator  of  a  common  fraction  may  be  any 
number,  but  the  denominator  of  a  decimal  fraction  must  be  10,- 100, 
or  1000,  etc. 

140.  A  decimal  is  written  at  the  right  of  a  period  (.) 
called  the  Decimal  Point. 

Note.  —  It  is  not  customary  to  write  the  denominator  of  a  decimal. 
It  is  determined  by  the  position  of  the  decimal  point. 

141.  A  figure  at  the  right  of  a  decimal  point  is  called  a 
Decimal  Figure.  Tenths  are  written  like  dimes  with  one 
decimal  figure ;  thus,  ^^  =  .5.  Hundredths  are  written  like 
cents,  with  two  decimal  figures ;  thus,  jW  =  .25,  -j^  =  .07. 
Thousands  are  written  like  mills,  with  three  decimal  figures ; 
thus,  Jo%V  =  -125,  TitTr  =  -016,  tAi7  =  -004.  Ten-thou- 
sandths  require  four  decimal  figures ;  hundred-thousandths, 
five  ;  millionths,  six ;  etc. 

142.  Name  the  denominators  in  the  following :  .36 ;  .08 ; 
.294;  .1406:  .0001;  .263402. 

PViQ-ncTA     fr»     rlppimnlQ*  25.        125.        10063.      3  6 . 

108 


TiAFr-  -^ 


DECIMAL   FRACTIONS.  109 

^     143.    A  Mixed  Decimal  is  an  integer  and  a  decimal;  as, 
16.04. 

144.    To  read  a  decimal. 

Rule.  — Read   the  decimal   as  an  integer,   and  give  it  the 
denomination  of  the  right-hand  figure. 

Read  the  following  numbers : 


1. 

.7 

8. 

.0000054 

15. 

235.850062 

2. 

.07 

9. 

35.18006 

16. 

100.000104 

3. 

.007 

10. 

.0005 

17. 

9.1632002 

4. 

.700 

11. 

.500 

18. 

3543.4536982 

5. 

.03065 

12. 

4.98625 

19. 

30.3303303 

6. 

.16984 

13. 

38694.06 

20. 

303.303303 

7. 

.10016 

14. 

9.98463004  • 

21. 

9.999999 

145.    To  write  a  decimal. 

\l  Rule.—  Write  the  numerator, prefixing  ciphers  when  necessary 
to  express  the  denominator,  and  place  the  point  at  the  left. 

Note.  —  There  must  be  as  many  decimal  places  in  the  decimal  as 
there  are  ciphers  in  the  denominator. 

Express  decimally : 
•    22.    Four   tenths.      Seventeen   hundredths.      Five    hun- 
dredths.     Three   hundred  twenty-five  thousandths.      Five 
thousandths.      Fifteen   thousandths.      Nineteen  and  seven 
hundred  twenty-four  thousandths. 

23.  Seven  thousand  five  hundred  four  ten-thousandths. 
Sixteen,  and  125  ten-thousandths.  Six  ten-thousandths. 
Five  thousand  ten-thousandths. 

24.  Seventeen  thousand  two  hundred  eleven  hundred- 
thousandths.  Four  hundred-thousandths.  Fifteen  hun- 
dred-thousandths. Eighteen,  and  two  hundred  sixteen 
hundred-thousandths.  One  hundred  twelve  hundred-thou- 
sandths. 


110 


DECIMAL   FRACTIONS. 


25.  Twenty-nine  hundredths.  Twenty-nine  thousandths. 
Twenty-nine  ten-thousandths.  Twenty-nine  hundred-thou- 
sandths. One  and  one  tenth.  One  and  one  hundredth. 
One  and  one  thousandth.  One  and  one  ten-thousandth. 
One  and  one  hundred-thousandth. 

26.  324  and  one  hundred  twenty-six  millionths.  4582 
and  36242  hundred-thousandths.  Seventeen  millionths. 
Five  hundred-thousandths.  Twenty-four,  and  three  thou- 
sand four  hundred  six  ten-millionths. 

27.  10  millionths.  824  ten-thousandths.  31  hundredths. 
216  hundred-thousandths.  7846  hundred-million ths.  Four 
and  15  hundred-thousandths. 


28. 

A 

32. 

ToVAiy 

36. 

t\ 

40. 

iTFOOOUiy 

29. 

1  5 
100 

33, 

ylttfio 

37. 

riTT 

41. 

nn 

30. 

T%\\ 

34. 

To'G^^'SU'U 

38. 

500A 

42. 

TTTO^OT 

31. 

tWA 

35. 

I^tAtt 

39. 

TWOTT 

43. 

TOOlJTJ 

REDUCTION  OF  DECIMALS. 

^       146.    Principles. — Ciphers  annexed  to  decimals  do  not 
change  their  value. 

V       For  each  cipher  prefixed  to  a  decimal,  the  value  is  dimin- 
ished tenfold. 

The  denominator  of  a  decimal  when  expressed  is  always 
1  with  as  many  ciphers  as  there  are  decimal  places  in  the 
decimal. 


^       147.    To  reduce  two  or  more  decimals  to  a  Common  Denomi- 
nator. 

V  Rule.  —  Aymex  ciphers  so  that  each   decimal  will  have  the 
same  number  of  decimal  figures. 


BEDUCTION    OF   DECIMALS.  Ill 

148.  Reduce  to  a  eommon  denominator: 

44.  .5,  .017,  .1256,  .000155,  29.803. 

45.  .80062,  305.24,  70.5,  3.85263. 

46.  .1,  .0001,  1000.001,  1.0100385. 

47.  .26,  .13682,  9.4,  25,  8.63521. 

149.  Reduce  .375  to  a  common  fraction. 

V    .375  as  a  common  fraction  is  xVoV     ^^^^  ^^  lowest  terms 

_  3 

—   8- 

s/ Rule. —  Write   the   numerator,  omitting   the  point.     Supply 
the  denominator,  and  reduce  to  lowest  terms. 

Reduce  to  common  fractions : 


48.    1.24 

53.    .32 

58.    16.144 

49.    .16 

54.    .113 

59.   28.3695 

50.    .325 

55.    .7282 

60.   34.000010 

51.   .098 

56.    2.25 

61.   25.0000100 

52.    .875 

57.    .2425 

62.    1084.0025 

150.    63. 

Red 

uce  37i  to  a  common 

fraction. 

Solution.  - 

100 

-fo=--^- 

Ans. 

64.    .12-1 

67.    .161 

70.    .87| 

65.    .06J 

68.    .33i 

71.    .661 

66.    .621 

69.    .831 

72.    .367 

151.   To  reduce 

a  common  fraction  to 

a  Decimal. 

Reduce  f 

to  a  ( 

iecimal. 

1  =  3  times  \.  3  =  (3.0),  30  tenths.  \  of  3.0  =  (.7), 
7  tenths,  and  2  tenths  remainder.  2  tenths  =  20  hun- 
dredths,    i  of  .20  =  .05.     Hence  |  =  .7  +  .05  =  .75. 


112  DECIMAL  FRACTIONS. 

\J Rule.  —  Annex  decimal  ciphers  to  the  numerator,  and  divide 
by  the  denominator.  Point  off  from  the  right  of  the 
quotient  as  many  places  as  there  are  ciphers  annexed. 

Notes.  —  A  decimal  cipher  is  a  cipher  at  the  right  of  the  decimal 
point.  If  there  are  not  enough  figures  in  the  quotient,  prefix  ciphers. 
The  division  will  not  always  be  exact.  In  such  cases  write  the 
remainder  over  the  divisor  as  a  common,  fraction,  or  place  the  sign 
+  after  the  decimal  to  show  that  the  result  is  incomplete.  Thus, 
|=.142f  or  .142+. 

162.   Reduce  to  decimals  : 


73. 

t 

77. 

A 

81. 

f 

85. 

t\ 

89. 

66| 

74. 

f 

78. 

1 

82. 

^4 

86. 

« 

90. 

25.121 

75. 

1 

79. 

f 

83. 

i 

87. 

12i 

91. 

161 

76. 

1 

80. 

i 

84. 

t\ 

88. 

331 

92. 

16.251 

ADDITION. 

153.   Add  .35,  4.375,  28.3065. 

.35  Rule.  —  Write  the  numbers  so  that  decimal  points 

4.375  stand  in  a  column.     Add,  as  in   integers,  and 

28.3065  place  the  point  in  the  sum  directly  under  the 

33.0315  points  above. 

Find  the  sum : 

93.         24.36  94.       38,28006  95.       1.186 

1.358                         1.005  .285 

.004                         2.16  .003 

1632.1                       1873.148^  203. 

96.  .175  4-1.75  +  17.5  +  175. +1750. 

97.  145.  +  14.5  +  1.45  +  .145  +  .0145. 

98.  32.58  +  28963.1  +  287.531  +  76398.9341. 

99.  1.  +  .1  +  .01  +  .001  + 100  +  10.  +  10.1  +  100.001. 


SUBTRACTION.  113 

100.  1.923  +  .008  ^  251.47  +  1.961  +  0.0543  +  .006  + 
18.7. 

101.  Add  750.3521,  698.42001,  .005321,  3.5,  749.006984, 
36950.06,  875.942,  286.753. 

102.  Add  5  tenths;  8063  millionths;  25  hundred-thou- 
sandths; 48  thousandths;  17  millionths;  95  ten-millionths ; 
5,  and  5  hundred-thousandths ;  17  ten-thousandths. 

103.  Add  24|,  17i,  .0058,  7^,  9J^. 


SUBTRACTION. 

154.  Rule.  —  Wi'ite  the  riumbers  so  that  the  decimal  point 
of  the  subtrahend  stajids  directly  under  the  decimal 
point  in  the  minuend.  Subtract  as  in  integers,  and 
place  the  point  directly  under  the  points  above. 

Note.  —  It  is  sometimes  convenient  to  give  the  decimals  the  same 
denominator  by  annexing  ciphers. 

104.    From  6.008     105.  38.  106.  26.34  107.  16.2600 

Take  3.154  .356  1.28983  1.0001 


108.  32.90596 -.75  114.    .00011  - .000011 

109.  9.5-3.35006  115.    10 -.1 +  .0001 

110.  856.2-8.562  116.    8.75 -h  .95  +  .125 

111.  .1 -.00001  117.    16-.00001  + 27.69852 

112.  1000 -.001  118.    2.5 -  .09  + 1.85 - 1.283 

113.  20 -.00205  119.    83.1  -  8.31  4- .831 

120.  From  one  thousand  take  five  thousandths. 

121.  Take  17  hundred-thousandths  from  1.2. 

122.  From  8.5  take  eighty-four  hundredths. 


114  DECIMAL   FRACTIONS. 

123.  Find  the  sum  of  500  thousandths  and  5  hundred- 
thousandths  and  from  it  subtract  ^^. 

124.  From  17.371  take  14.16i 

125.  Find  the  difference  between  f-^-^-^  and  yfft^. 

126.  From  10  take  J^ ;  j^^-,  4.98;  1.05. 

127.  From  one  million  and  one  millionth  take  one  tenth. 

128.  From  1  tenth  take  1  millionth. 

129.  Which  is  the  greater  and  how  much,  one  tenth  or 
100  thousandths  ? 

130.  Prove  that  i  and  .500  are  equal. 


M  UL  TIP  Lie  A  TI  ON. 

155.  Every  decimal  equals  a  corresponding  common 
fraction,  and  for  each  cipher  in  its  denominator  there  is  a 
decimal  figure  in  the  decimal  fraction. 

TTO"  ^  i^  —  tMu-     (Three  ciphers  in  the  denominator.) 
.05  X  .3  =  .015.     (Three  decimal  places  in  the  decimal.) 

Rule.  —  Multiply  as  in  iritegers,  and  give  to  the  product  as 
many  decimal  figures  as  there  are  in  both  multiplier  and 
multiplicand. 

Note.  —  If  there  are  not  figures  enough,  prefix  ciphers. 
Ciphers  at  the   right  of   a, decimal  have  no  value,  and  may  be 
omitted. 

Find  the  products : 

1.  .38  X  1.6  7.  .296  x  124 

2.  .015  X  .05  8.  1.001  X  1.01 

3.  1\  X  3.4  9.  13.33  X  1.3 

4.  50  X  .304  10.  25.863  x  4J 

5.  2.65  X  .104  11.  1.04  x  ^ 

6.  257  X. 354  12.  327f  x  4| 


MULTIPLICATION.  115 

13.  58.42  X  20.06  17.  .001542  x  .0052 

14.  .0001  X  1000  18.  26  X  36.82 

15.  .325  xl2|-  19.  2.84  x  3J 

16.  .333  X  .333  20.  11.11  x  100 

156.    To  multiply  by  lo,  lOO,  looo,  etc. 

21.    Multiply  1.265  by  100. 

1.625  Remove  the  point  one  place  to  the  right  for 

100        each  cipher  in  the  multiplier. 


126.500            Do  not  write  the 

multiplier. 

Oral. 

22.    3689.25  x  10 

27.    .5  X  100 

23.    38.6422  x  100 

28.    .5  X  1000^ 

24.    269.8342  x  1000 

29.    384.2  X  10 

25.    100  X  23.85 

30.    .3659  X  100 

26.    1000  X  1.52 

31.    .1000  X  .01 

157.  To  multiply  by  200,  remove  the  point  to  the  right 
and  multiply  by  2. 

Oral. 

32.  86.44  X  200  35.  750.5  x  5000 

33.  3.894  X  3000  36.  1.892  x  2000 

34.  88.42  X  20  37.  156.2  x  200 

158.  Written. 

38.  Find  the  product  of  1  thousand  by  one  thousandth. 
1  million  by  one  millionth. 

39.  Multiply  700  thousandths  by  7  hundred-thousandths. 

40.  Multiply  the  sum  of  2  millionths  and  10  thousandths 
by  their  difference. 

41.  Multiply  together  .35,  18.5,  28.004. 


116  DECIMAL   FKACTIONS. 


DIVISION. 


159.  Since  in  multiplication  there  are  as  many  decimal 
places  in  the  product  as  in  both  multiplier  and  multiplicand, 
in  division  the  quotient  must  have  as  many  places  as  the 
number  of  places  in  the  dividend  exceeds  those  in  the 
divisor. 

1.  Divide  12.685  by  .5. 

K\-j2fjcK  Solution. — Since  there  are  three  decimal  places 

or'orr  "      i"  the  dividend  and  one  in  the  divisor,  there  must  be 
two  m  the  quotient. 

iZizie. »— 1.    In  all  cases  divide  as  in  integers,  then  place  the 
decifial  point. 
0 

2.  Divide  399.552  by  192. 

1Q9\QQQ^W      ■^w-'^'  —  2-    When  the  divisor  is  an  integer, 

^^ooj^'  place  the  point  in  the  quotient  directly 

TbbE  ^^^^  ^^^  point  in  the  dividend  in  long 

1536  division  (directly  under   in  short  divi- 

192  sion).     Prove  by  multiplying  divisor  by 

192   V         quotient. 

Principle.  —  Multiplying  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 

3."  Divide  28.78884  by  1.25. 

23.031  + 
1.25')28.78'884  Rule.  —  3.    When    the   divisor  contains 

250  decimal  figures,  move   the  point  in 

378  both  divisor  and  dividend  as  many 

^^^  places  to  the  right  as  there  are  deci- 

mal places   in  the  divisor  (this,  in 


388 


§Z5_  Ex.  3,  multiplies  both  by  100),  then 


134 
125 


place  the  point  in  the  quotient  as  if 


'  the  divisor  were  an  integer. 


DIVISION.  117 

Note  1.  — The  new  points  may  be  placed  on  a  line  with  the  tops 
of  the  figures,  and  the  original  points  may  stand  to  preserve  the 
reading  of  the  decimals. 

Note  2.  —  If  the  quotient  does  not  have  a  sufficient  number  of 
figures,  prefix  ciphers. 

Note  3.  —  Before  commencing  to  divide,  see  that  there  are  at  least 
as  many  decimal  places  in  the  dividend  as  in  the  divisor. 

Note  4.  —  If  there  is  a  remainder  after  all  the  figures  of  the  divi- 
dend are  used,  annex  decimal  ciphers  and  continue  the  division. 

Note  5.  —  It  is  not  usually  necessary  to  have  more  than  four  deci- 
mal figures  in  the  quotient. 


Find  the  quotients : 

1.    .288 -.64 

11. 

315.432  -  .132 

2.    .36-600 

12. 

1.5906  -  241 

3.    144 -.12 

13. 

36.25  - 1.25 

4.    .25 -.2500 

14. 

75  -  .0125 

5.    .12-30 

15. 

125  -  .12^ 

6.    .96H-.08 

16. 

25  -  .25 

7.   384.526-1.16 

17. 

.25  -  25 

8.    1440 -.0018 

18. 

1000  -  .001 

9.    1.225-4.9 

19. 

.001  - 1000 

10.    9.156-12 

20. 

18.65  - 100 

160.    To  divide  by  lo,  loo,  looo,  etc.,  remove  the  point  one 
place  to  the  left  for  each  cipher  in  the  divisor.  , 

Oral. 

21.  38.64-10  25.   3.91 -H- 1000 

22.  .5-^10  26.    1.155  h- 100 

23.  558-^100  27.   398.42-1000 

24.  1684.32-1000  28.    2.46-200 

Note.  —  To  divide  by  200,  remove  the  point  to  the  left,  and  divide 
by  2. 

29.  386.54-2000  31.    865.45-5000 

30.  38.28-5-400  32.    2.5-500 


118  '   DECIMAL   FRACTIONS. 

PARTS  OF   lOO  OB  lOOO. 

161.  1.   What  part  of  100  is  121  ?     25?     33i? 

2.  What  part  of  1000  is  125  ?     250  ?     333|  ? 

3.  How  much  is  J  of  100  ?     Of  1000  ? 

4.  How  much  is  i  of  100  ?     Of  1000  ? 

5.  Find  J  of  100.     Of  1000. 

»  6.    How  much  is  25  times  24  ? 

Solution.  — 100  times  24  =  2400. 

25  times  24  =  ^  as  much  as  100  times  24,  =  600. 

162.  To  multiply  by  25,  annex  two  ciphers,  and  take  J  of 
the  result. 

7.  Tell  how  to  multiply  by  33i ;  by  12i;  by  250;  by 
125 ;  by  3331 

Oral. 

8.  36  X  25  11.   444  x  25  14.    3331  x  30 

9.  48  X  121  12.   320  X  33J  15.    168  x  250 
10.    24  X  33i             13.    125  X  80  16.    12J  x  48 

17.  What  cost  650  oysters  at  50  cents  a  hundred  ? 
Solution.  —  650  h-  100  =  6.50  hundred. 

$.50x6.50=? 

18.  What  will  be  the  cost  of  3850  laths  at  40  cents  a 
hundred  ? 

19.  What  is  the  freight  on  685  pounds  of  baggage  at 
$  1.10  per  100  lb. 

Note.  —  C.  means  100  ;  M.,  1000. 

20.  What  is  the  cost  of  4862  ft.  of  pine  lumber  at  $  30 
per  M.  ? 

21.  Find  the  cost  of  38,586  bricks  at  $  8.25  a  thousand. 

22.  What  will  583  heads  of  cabbage  cost  at  $  3.50  a  hun- 
dred ? 


WRITTEN    EXEKCISES^  ..  119 

23.  At  $3.50  a  thousand,  what  will  be  the  cost  of  7800 
shingles  ? 

24.  At  $8.25  per  C,  what  will  be  the  cost  of  2864  lb.  of 
dried  fish  ? 

25.  At  $  50  per  M.,  what  will  be  the  cost  of  3865  feet  of 
cherry  lumber  ? 

26.  What  is  the  cost  of  laying  5890  bricks  at  $  9.00  a 
thousand  ? 

To  find  the  cost  of  merchandise  sold  by  the  ton,  divide 
the  price  by  2  and  proceed  as  above. 

27.  Three  loads  of  hay  weigh  7894  lb.  What  will  the 
hay  bring  at  $  12  a  ton  ? 

Note.  — 1000  lb.  will  cost  ^  of  $  12  =  $  6.     $  6  x  7.894  =  ? 

28.  What  cost  48986  lb.  of  railroad  iron  at  $  35  a  ton  ? 

29.  Four  loads  of  coal  weigh  respectively  3896  lb.,  3524 
lb.,  4106  lb.,  and  3123  lb.  What  is  the  cost  of  the  coal 
at  $  4.82  a  ton. 

ALIQUOT  PARTS  OF  $1.00. 

163.  The  Aliquot  Parts  of  a  number  are  the  numbers 
which  are  exactly  contained  in  it. 

The  aliquot  parts  of  100  are  5,  20,  12-i-,  16|,  33|,  etc. 

164.  The  aliquot  parts  of  $  1,  commonly  used,  are  as 
follows : 

61  cents  =  $J^  25    cents  =  $  J 

8i  cents  =  $^  33^  cents  =  $  J 

12|  cents  =  $  |-  50    cents  =  $  J 

16|  cents  =  f  J- 

1.    What  is  the  cost  of  69  books  at  16|^  each  ? 

Solution.  —69  books  will  cost  69  times  16|j?,  or69x$J  =  $^  = 
$11.60.     Ans. 


120  DECIMAL  FRACTIONS. 

165.  Oral. 

Multiply : 

2.  33 J  cents  by  36  5.    25  cents  by  40 

3.  121  cents  by  24  6.    75  cents  by  4 

4.  ej  cents  by  32 

Note.  — ^  means  cents,  lb.,  pounds,  and  yd.,  yards. 

7.  What  is  the  cost  of : 

48  lb.  of  bacon  at  121^  a  pound  ? 
80  hand  balls  at  50^  each  ? 
36  yd.  of  ribbon  at  33^^  a  yard  ? 
80  lb.  of  candy  at  25^  a  pound  ? 

166.  Written. 

8.  Find  the  cost  of  the  following : 

66  lb.  of  pork  at  121^, 
148  lb.  of  veal  at  16f^, 
48  boxes  of  strawberries  at  25^, 
48  lb.  of  honey  at  25^, 
64  bars  of  soap  at  6^^, 
60  doz.  of  eggs  at  16|^. 
Find  the  cost  of : 

9.  1580  lb.  of  sugar  at  6J^  a  pound. 

10.  500  books  at  25^  each. 

11.  16  yd.  of  dress-goods  at  33^^  a  yard. 

12.  At  25^  a  pound,  how  many  pounds  of  butter  can  be 
bought  for  $  8.00  ? 

Solution.  —  As  many  pounds  as  25j?  or  $  |  is  contained  times  in 

167.  Oral. 

Divide : 

13.  $  5  by  331^  16.    $  3  by  8J^ 

14.  $6  by  6i^  17.    ^4  by  25^ 

15.  $  9  by  12^^  18.    f  4  by  66|^ 


REVIEW   OF   DECIMALS.  121 

19.  At  25^  each,  how  many  hats  can  be  bought  for  $  6  ? 

20.  At  $  J  a  pound,  how  many  pounds  of  cheese  can  be 
bought  for  $  6  ? 

21.  At  33 J^  a  yard,  how  many  yards  of  linen  can  be 
bought  for  i  10  ? 

168.  Written. 

22.  At  75^  a  bushel,  how  many  bushels  of  barley  can  be 
bought  for  $  125  ? 

23.  When  butter  is  25P  a  pound,  how  many  pounds  can 
I  buy  for  $50? 

24.  How  many  dozen  eggs  at  16J  cents  a  dozen  can  be 
bought  for  $  38  ? 

25.  At  12|^  cents  a  quart,  how  many  quarts  of  nuts  can 
be  bought  for  $  10  ? 

REVIEW  OF  DECIMALS, 

169.  1.   Tell  how  to  locate  the  decimal  point  in  any  sum. 
In  any  remainder.     In  any  product.     In  any  quotient. 

2.  In  the  number  777,  what  is  the  local  value  of  the  7  at 
the  right  ?     The  second  7  ?     The  left-hand  7  ? 

3.  Upon  what  does  the  value  of  any  figure  depend  ? 

4.  In  the  decimal  .777,  what  is  the  value  of  the  first  7 
at  the  right  ?     The  second  7  ?     The  third  7  ? 

5.  What  is  the  effect  of  removing  an  integral  figure  one 
place  to  the  right  ?     A  decimal  figure  ? 

6.  What  is  the  effect  of  removing  an  integral  figure 
one  place  to  the  left  ?     A  decimal  figure  ? 


122  DECIMAL   FRACTIONS. 


Eead: 

7.   .0001 

.00196 

4.3 

.0006 

.02789 

71.86 

.0014 

.52000 

329.400 

.0282 

.050798 

'   1.001 

.5897 

.725386 

200.3278 

.00001 

.500001 

579000.00005 

.00027 

.000829 

437.050609 

Copy  and  write  decimally  : 

8.  1  tenth;  24  hundredths ;  379  thousandths ;  1000  ten- 
thousandths  ;  85  hundred-thousandths ;  20079  millionths. 

9.  One  thousand  six,  and  five  hundred  two  millionths. 

10.  Three  hundred  fifteen  thousand  one,  and  eleven  ten- 
thousandths  ;  thirty-eight,  and  seven  thousandths ;  8  mil- 
lion 270  thousand  942,  and  5  thousandths  ;  seventeen  tenths. 

11.  Four  hundred  21,  and  5  ten-thousandths ;  1  thousand 
27,  and  27  hundredths;  ninety -nine,  and  ninety-nine  ten- 
millionths. 

Write  without  the  denominator : 

42t%,  78t%Vt5,  2003-VA^^. 

13.    Change  to  common  fractions  in  lowest  terms : 

.028,  .0015,  .2175,  .000048,   .00075,  .45,  .8,  .75,  8.9375, 

91.16,  4001.645,  9.156575. 

Change  to  equivalent  decimals : 

14-  h  h  -h,  H,  A.  I.  20H,  8ji^,  4J5,  losm. 

Change  to  common  fractions,  then  to  simple  decimals : 
15.    .1^,  .07i,  .18f,  .mi,  .121,  .08i,  .221,  .045^%,  ,37J, 
,381,  .541,  .000051   j8|,  .38J 


REVIEW   OF   DECIMALS.  123 

Reduce  to  a  common  denominator  and  add : 

16.  50.06,  367.41,  200.200,  .12304,  40.0056,  7.5620, 
.096071. 

17.  1301.6,  904.02,  .547,  .0009,  .00001,  218.94,  203.410, 
1000,  .01. 

18.  100.101,  82.4,  401.009,  .00038,  60702,  10.10, 
574.68139. 

19.  5.628,  850.002,  9.00256,  37.0005,  724.6811,  3759, 
7000.0036,  2.25. 

20.  $11.78,  $347,  $5.06,'  $218,  $20.07,  $42.0244, 
$7,104,  $37,625. 

21.  4.76,  .390,  .0915,  .00207,  841,  63.2,  .00234,  1.43, 
.00536. 

22.  .00908,  .0371,  24.5,  7.03,  .0127,  354,  .000781,  .0436, 
20.7354. 

Subtraction : 

23.  5.74-3.23  =  ?  26.    367-1.52  =  ? 

24.  .876 -.343  =  ?  27.    200  -  .02  =  ? 

25.  67.5-41.5  =  ? 

28.    Which  is  greater,  |  or  4  tenths  ? 
.    29.    How  much  more  is  $  20  than  $  17.84  ? 

30.  Erom  two  million  take  two  millionths. 

31.  I  bought  4  farms:  one  contained  19.368  acres;  one, 
27.96  acres;  one,  473.0008  acres;  and  one,  73.7561  acres. 
I  sold  300.25  acres.     How  much  land  had  I  left  ? 

32.  From  1  inch  take  one  ten-thousandth  of  an  inch. 

Multiply  : 

33.  7.945  by  .3  37.  7.853  by  23.16 

34.  350  by  .42  38.  1.36  x  20.04  =  ? 

35.  One  tenth  by  one  hundredth.     39.    27.27  x  4.0004  =  ? 

36.  25  units  by  25  tenths. 


124  DECIMAL  FRACTIONS. 

40.  If  wheat  is  worth  $  .38  a  bushel,  what  will  117.75 
bushels  cost? 

41.  Apples  sell  for  $1.28  a  bushel.     How  much  money 
will  24  barrels  bring,  each  containing  2i  bu.  ? 

42.  Find  the  cost  of  3.325  lb.  of  butter  at  18.75  cents  a 
pound. 

43.  What  will  6|  yd.  of  broadcloth  cost  at  $  1.375  a  yd.  ? 

44.  A  boy  paid  $  .125  a  dozen  for  1.75  dozen  eggs.     What 
did  they  cost  him  ? 

45.  3.64  X  .0002  X  1.756  x  4.004  =  ? 

Divide : 

46.  1738.89  by  .00417.  52.  42.475681  by  .29. 

47.  1237.6  by  26.  53.  40.20  by  .000012. 

48.  36.11  by  .021.  54.  $  302.03  by  200. 

49.  2.38  by  .17.  55.  64.64006  by  .002. 

50.  36.82  by  .0003.  56.  12.9643  by  18.4. 

51.  437.96  by  2.8.  57.  759.806  by  90.3. 

58.  16|  +  3.06  - 1  +  .002  -  2.1  +  .03  - 1  +  .OOi  =  ? 

59.  ^  +  3  _  .65  +  .5  +  J  -  i  +  3.14  =  ? 

60.  Find  the  product  of  .003  multiplied  by  .06,  and  divide 
it  .by  3. 

61.  A  certain  decimal  divided  by  1000  is  35.002.     What 
is  one  fifteenth  of  the  decimal  ? 

62.  The  sum  of  two  numbers  is  306.52 ;  one  of  them  is 
100.     What  is  the  other  ? 

63.  A  man  spent  $450,  which  was  .125  of  his  money. 
How  much  money  had  he  ? 

64.  Mr.  A.  bought  a  cow  for  $  45,  which  was  .375  of  what 
he  paid  for  a  horse.     How  much  did  he  pay  for  the  horse  ? 

65.  John  spent  .75  of  his  money  for  a  book  and  had  $  .50 
left.     How  much  had  he  at  first  ? 


ACCOUNTS   AND   BILLS. 


170.  An  Account  is  a  record  of  indebtedness  for  articles 
bought  or  sold,  cash  paid  or  received,  or  services  rendered. 

171.  A  Debtor  is  a  person  who  owes  a  debt. 

172.  A  Creditor  is  a  person  to  whom  a  debt  is  owed. 

173.  A  Bill  is  a  written  statement  of  a  debtor's  account, 
made  by  the  creditor. 

174.  A  Receipt  is  a  creditor's  written  acknowledgment 
that  he  has  received  payment  of  part  or  all  of  a  debt. 

175.  A  bill  is  receipted  when  its  payment  is  acknowl- 
edged in  writing,  by  the  creditor,  or  by  some  authorized 
person. 

Note.  — The  sign  @  is  for  at.  Dr.  is  for  debtor,  and  Cr.  for 
creditor. 

1. 


BILL  FORMS. 


James  P.  Barnes, 


Chicago,  July  1,  1902. 
Bought  of  Dey  Bros.  &  Co. 


50  yd.  Brussels  Carpet   @ 
24   "    Oil  Cloth  " 

4  doz.  pair  Merino  Hose  " 
2  Willow  Chairs  " 


$i 

15 

35 

f  : 

8 

50 

4 

50 

1$ 

125 


126 


ACCOUNTS   AND   BILLS. 


RECEIPTED  BHili  WITH  CREDITS. 

Rochester,  jST.  Y.,  Jan.  S,  1896, 
Mrs.  Johx  F.  White, 

1895 


To  Burke  &  White,  Dr. 


Nov. 

6 

11 

6 

li 

18 

Dec. 

11 

« 

15 

ii 

19 

Nov. 

18 

Dec. 

28 

4  lb.  Coffee 
28  lb.  Sugar 

5  gal.  Molasses 
18  lb.  Rice 

2  bbl.  Potatoes 
28  lb.  Butter 


@ 


Cash 


Balance  due, 


27 
5% 
60 

7% 
80 
21 

50 

75 


$ 


Received  payment,  Jan.  15,  1896, 

BuKKE  &  White, 

By  John  R.  Pierce. 


FORM  OV  A  RECEIPTED  BIIX. 

New  York,  June  SO,  1896. 
Jerome  A.  Phelps, 

In  account  with  D.  0.  Potter  &  Co. 


May 

H 

12  bbl.  Flour 

@ 

$6 

50 

$ 

K 

U 

6  tubs  Butter,  684  l^- 

ii 

24 

June 

10 

5  bbl.  Beef 

u 

25 

28 

u 

25 

450  lb.  Ham 

u 

9% 

Received  payment, 

D.  O.  Potter  &  Co. 


WRITTEN    EXERCISES.  127 

4.  Mr.  John  Q.  Adams  buys  of  D.  McCarthy  &  Co. : 

14  pounds  of  coffee  at  27  cents  a  pound, 
28  pounds  of  sugar  at  5^  cents  a  pound, 

15  gallons  of  molasses  at  60  cents  a  gallon, 

16  pounds  of  rice  at  SJ  cents  a  pound. 
Make  out  the  bill. 

5.  James  Smith,  farmer,  sold  Kichard  Dunn,  grocer,  the 
following : 

16  barrels  of  potatoes  at  $1.80  a  bbl., 

12  tons  of  hay  at  $  16  a  ton, 

13  cords  of  wood  at  $  4  a  cord, 

360  pounds  of  butter  at  24A  ^  a  pound. 
Make  a  receipted  bill.  , 

6.  Chicago,  Dec.  5,  1900.  Edward  Smith  sold  B.  M. 
Watson  65  yd.  Brussels  carpet  at  $  1.25 ;  24  yd.  oil  cloth  @ 
35^;  one  dozen  pair  of  merino  hose  @  $3.50;  2  willow 
chairs  @  $  4.50.#  Make  bill,  find  the  footing,  and  properly 
receipt  it. 

7.  Make  out  a  bill  of  groceries.  Foot  it,  and  receipt  it, 
with  F.  H.  Mead  as  creditor  and  Wm.  H.  Scott  as  debtor. 

To  THE  Teacher. — See  that  the  prevailing  prices  are  used,  and 
that  the  quantities  are  consistent. 

8.  J.  H.  Acker  bought  of  H.  A.  Strong  of  San  Francisco, 
the  following  articles :  15  bbl.  flour  @  $  8.00 ;  6  tubs  of 
butter,  120  pounds  in  a  tub,  at  24  cents  a  pound ;  5  barrels 
of  beef,  200  lb.  to  the  barrel,  at  5^;  25  sacks  of  flour  @ 
95^;  450  pounds  of  ham  at  11^^.  Make  out  bill,  and 
receipt  it. 

9.  Make  out  a  bill  of  hardware,  another  of  groceries, 
and  another  of  dry  goods. 

,    10.    Make  out  a  bill  of  goods  bought  at  a  shoe  store ;  at  a 


128  MISCELLANEOUS. 

MISCELLANEOUS. 

176.  M.  I  have  four  pieces  of  broadcloth.  The  first  con- 
tains 13.7642  yd.;  the  second,  22.008  yd.;  the  third,  15.027 
y(L;  and  the  fourth,  19.255  yd.     How  many  yards  in  all  ? 

"•  2.    From  a  piece  of  ribbon  containing  103f  yd.,  73|  yd. 
were  sold.     How  many  yards  were  left  ? 

3.  How  many  yards  of  muslin  at  $  .121  a  yard  will  it 
take  for  4  pair  of  curtains,  if  each  curtain  contains  3.375  yd.  ? 

4.  I  have  14.735  yd.  of  lace,  and  desire  to  cut  it  into 
seven  equal  strips.     How  much  will  there  be  in  each  strip  ? 

5.  What  will  be  the  cost  of  a  hat  at  $  7.50,  a  pair  of 
gloves  at  $  1.13,  a  veil  at  $  1.25,  and  a  parasol  at  $  3.375  ? 

6.  Arrange  the  following  articles  in  the  form  of  a  bill : 
7  qt.  of  molasses  at  $.15  a  qt.,  |  bu.  of  apples  at  $1.28  a 
bushel,  30  lb.  of  sugar  at  $.08^  a  pound,  and  12  bu.  of 
potatoes  at  $  .29  a  bushel. 

7.  A  grocer  bought  three  bunches  of  bananas  at  $  1.54 
a  bunch.  The  first  bunch  contained  73  bananas,  the  second 
54,  and  the  third  97.  He  sold  them  all  at  30^  a  dozen. 
Did  he  gain  or  lose,  and  how  much  ? 

8.  The  first  year  in  business  a  grocer  made  $  2374.68, 
the  second  $  1529.47,  and  in  the  third  year  he  lost  $  300. 
His  expense  each  year  averaged  $928.45;  how  much  money 
had  he  gained  at  the  end  of  three  years  ? 

9.  What  will  9  barrels  of  flour  cost,  if  28  barrels  cost 
$173.60? 

10.  I  bought  437  heads  of  lettuce  at  $  5  a  hundred,  and 
sold  them  at  $  .08  a  head.     What  was  my  gain  ? 

Find  the  cost  of ; 

11.  6824  1b.  of  coal  at  $4.68  a  ton. 

12.  2384  lb.  of  coal  at  $  5.67  a  ton. 

13.  8972  ft.  of  lumber  at  $  35.40  a  thousand. 


17.    What  part  of  4.50  is  3.33J  ? 


MISCELLANEOUS.  129 

14.  6854  lb.  of  hay  at  $  16.50  a  ton. 

15.  4836  bricks  at  I?  9.45  per  M. 

16.  895  ft.  of  lumber  at  $  19.75  per  M. 

9 

18.  What  part  of  3.625  is  1.5  ? 

19.  What  part  of  6.2  is  3.25  ? 

20.  1.1  is  what  part  of  7.4  ? 

21.  A  father  left  his  son  $24,000,  which  was  .375  of  his 
estate.     What  was  the  value  of  the  estate  ? 

22.  Divide  26  by  2^,  and  multiply  the  result  by  17.345. 

23.  Divide  |  of  .375  by  f  of  |  of  .298. 

24.  The  product  of  three  numbers  is  167.7.  Two  of  the 
numbers  are  3.25  and  5.16.     What  is  the  other  ? 

25.  What  number  divided  by  2.86  equals  .34  ? 

26.  What  number  diminished  by  38.64  leaves  .356  ? 

27.  A  man  bought  8.5  yd.  of  cloth  at  $3.33J  a  yard, 
12.4  yd.  at  $  2.75,  18^  yd.  at  $4,375,  and  24f  yd.  at  $  2.875. 
How  many  bushels  of  corn  at  43|  cents  a  bushel  will  pay 
for  the  cloth  ? 

28.  .5  of  a  number  exceeds  .45  of  it  by  20.  What  is  the 
number  ? 

Solution.  — .5  —  .45  =  .05.  Now  the  question  is,  20  is  .05  of  what  ? 
20  -f-  .05  =  400. 

29.  At  85  j^  a  yard,  how  many  yards  of  cloth  can  be  pur- 
chased for  $29.75? 

30.  Divide  $  785  among  A,  B,  and  C,  so  that  C  will  have 
$  185  more  than  each  of  the  others. 

31     1_ .0045 

"    .05      .4  X  .005  +  .002  x  .125 


130  MISCELLANEOUS. 

32.  What  part  of  .876  is  »31536  ? 

33.  If  .375  of  a  ton  of  coal  cost  $  1.25,  what  will  7.125 
tons  cost  ? 

34.  What  is  .3  of  a  number  when  .8  of  it  is  80  ? 

35.  How  many  thousandths  in  3  units  ? 

36.  How  many  thousandths  in  .1  ? 

37.  Express  \  of  one  hundredth  as  a  decimal. 

38.  The  salary  of  the  President  of  the  United  States  is 
$  50,000  a  year.     How  much  does  he  receive  per  day  ? 

33  i  of  4  5| 

39.  Divide  the  product  of  5  times  j|  plus  ^        •'^    by  ^• 

40.  Divide  2  of  il  by  J  of  ^. 

Find  the  cost  of  the  following : 

41.  3151  lb.  of  tea  at  $  .37i  a  pound, 
34f  lb.  of  coffee  at  $  .18f  a  pound, 
3105|  lb.  of  pork  at  $  .121  a  pound, 
30691  bu.  of  wheat  at  f  1.121  a  bushel, 
36|  doz.  of  eggs  at  $  .121  a  dozen, 

26|  yd.  of  sheeting  at  $  .07|  a  yard. 

42.  A  owns  f  of  a  farm  and  B  owns  the  remainder;  J  of 
the  difference  of  their  shares  is  worth  $  2400.  What  is  the 
value  of  the  farm  ? 

43.  Divide  $  3J  among  some  poor  children,  giving  each  J 
of  a  dollar.     W^hat  will  be  the  number  of  children  ? 

44.  Two  men  hire  a  pasture  for  $  25.  A  puts  in  8  horses 
and  B  12  horses.     How  much  should  each  pay  ? 

Note.  —  Both  have  put  in  20  horses.    A  must  pay  ^  and  B  i^  of  $  25. 

45.  Add  8  to  both  terms  of  the  fraction  ■^,  and  find  how 
much  you  have  increased  or  diminished  it. 


QUESTIONS.  131 

46.  Subtract  4  from  each,  term  of  the  fraction  ^,  and  find 
how  much  it  has  been  increased  or  diminished. 

47.  Find  the  least  common  multiple  of  28,  34,  42,  and  56. 

48.  Divide  the  least  common  multiple  of  240  and  600  by 
their  greatest  common  divisor. 

49.  Name  all  the  prime  numbers  between  75  and  100. 
The  odd  numbers. 

50.  Divide  the  product  of  21  x  11  x  6  x  26  x  10  by  the 
product  of  5  X  13  X  3  X  14  X  2.     Use  cancellation. 

51.  A  merchant  bought  10  casks  of  vinegar,  each  contain- 
ing 42  gallons,  at  20  cents  a  gajlon,  and  paid  for  them  in 
pieces  of  cloth,  each  containing  35  yards,  at  10  cents  a  yard. 
How  many  pieces  of  cloth  did  he  give  ? 

QUESTIONS. 

/    177.    l.^What  is  a  decimal  ?  mow  are  decimals  written  ? 
'    Why  are  they  called  decimals  ? 

2.  How  many  decimal  places  are  needed  to  write  ten- 
thousandths  ?  •  Millionths  ?  *  Hundredths  ? 

3.  ''What  is  the  first  place  at  the  right  of  the  decimal 
point?  "^'What  is  the  first  period  called ?*  The  second 
place  ?  ^  The  second  period  ? 

4. '  What  is  a  mixed  decimal  ? 

.  5.    What  must  the  denominator  of  a  decimal  be  ? 

6.   What  is  the  effect  of  removing  the  decimal  point  one 
place  to  the  right  ?  '  To  the  left  ?^  Two  places  to  the  right  ? 
^  Three  places  to  the  left  ? 

7."^  What  is  the  effect  of  annexing  a  cipher  to  an  integer  ? 
To  a  decimal?  '  Of  prefixing  a  cipher  to  an  integer  ?  ^  To  a 
decimal  ? 


132  MISCELLANEOUS. 

8.  How  do  we  reduce  decimals  to  common  fractions? 
Common  fractions  to  decimals  ? 

9.  Give  rules  for  adding,  subtracting,  multiplying,  and 
dividing  decimals. 

10.  How  do  we  locate  the  decimal  point  in  the  sum  ? 
In  the  remainder  ?      In  the  product  ?      In  the  quotient  ? 

11.  What  are  coins  ? 

12.  What  are  the  gold,  silver,  bronze,  and  nickel  coins 
used  in  the  United  States  ? 

13.  What  are  the  aliquot  parts  of  a  number  ?     What  are, 
the  aliquot  parts  of  ^  1  ?     Of  100  ?     Of  1,000  ? 

14.  What   is  a  bill?     An   account?     A  creditor?     A 
debtor?     Tell  how  to  receipt  a  bill. 

178.    1.    Define  unit,  number,  the  unit  of  a  number,  ab- 
stract number,  concrete  number,  li-ke  numbers. 

2.  Define  notation,  numeration,  Arabic  notation. 

3.  What  is  the  value  of  the  unit  figure  of  a  number? 
The  tens  ?     The  hundreds  ? 

4.  What  is  the  largest  number  which  can  be  expressed 
by  four  figures  ? 

5.  What  is  the  simple  value  of  a  figure  ?  The  local 
value  ? 

6.  What  name  is  given  to  the  first  period  to  the  right 
of  the  decimal  point?     The  second?     The  third? 

7.  What  is  addition?  What  kind  of  numbers  can  be 
added? 

8.  Define  subtraction,  minuend,  subtrahend,  remainder. 
What  is  a  proof  of  subtraction?  What  is  the  sign  of  sub- 
traction, and  where  placed? 


QUESTIONS.  .  133 

9.    What  is  a  parenthesis?      A  vinculum?      For  what 
are  they  used? 

10.  What  is  multiplication ?  The  multiplier?  The  mul- 
tiplicand?    The  product? 

11.  The  multiplier  and  the  multiplicand  are  what  of  the 
product  ? 

12.  What  is  the  sign  of  multiplication  and  how  is  it 
used  ?  Define  division,  divisor,  dividend,  quotient,  re- 
mainder. 

13.  What  is  the  sign  of  division,  and  how  is  it  used? 

14.  Express  the  division  of  12  by  8  in  as  many  ways  as 
you  can. 

15.  To  what  terms  in  multiplication  do  the  divisor,  quo- 
tient, and  dividend  correspond? 

16.  How  do  you  find  the  dividend  when  the  divisor,  quo- 
tient, and  remainder  are  given? 

17.  When  is  the  quotient  an  abstract  number? 

18.  When  the  quotient  and  dividend  are  like  numbers, 
what  kind  of  a  number  is  the  divisor? 

19.  How  can  we  divide  when  the  divisor  is  10?  100? 
1000?     When  the  divisor  is  20?   50?   300? 

20.  Multiplying  both  dividend  and  divisor  by  the  same 
number  affects  the  quotient  how? 

21.  Dividing  both  divisor  and  dividend  by  the  same 
number  affects  the  quotient  how? 

22.  Multiplying  the  dividend  affects  the  quotient  how? 
The  divisor?     Dividing  the  dividend?     The  divisor? 

23.  Define  exact  divisor,  factor,  prime  factor,  factoring. 

24.  How  can  you  find  the  prime  factors  of  a  number  ? 


134  .     Mli^rELTvAKEOITS. 

26.  Dej&ne  divisor.  Common  divisor.  The  greatest  com- 
mon divisor.  Give  the  riile  to  find  the  greatest  common 
divisor. 

26.  Define  multiple,  common  multiple,  least  common 
multiple.  Give  the  rule  for  finding  the  least  common 
multiple. 

179.  Define  fraction,  fractional  unit,  unit  of  a  fraction, 
denominator,  numerator,  terms  of  a  fraction,  common  frac- 
tion, integer,  proper  fraction,  improper  fraction,  mixed  num- 
ber, simple  fraction,  compound  fraction,  complex  fraction. 

What  is  the  value  of  a  fraction? 

State  the  principles  of  fractious. 
■    What  is  it  to  reduce  a  fraction  ? 

How  are  fractions  reduced  to  lowest  terms?  To  higher 
terms  ? 

How  can  an  improper  fraction  be  reduced  to  a  whole 
or  a  mixed  number?  A  whole  or  a  mixed  number  to  an 
improper  fraction? 

What  are  like  fractions?     Unlike  fractions? 

How  can  fractions  be  reduced  to  others  having  a  common 
denominator?     A  least  common  denominator? 

How  can  two  or  more  fractions  be  added? 

How  can  the  sum  of  fractions  be  found?  Mixed  num- 
bers? 

How  can  the  difference  of  fractions  be  found?  Mixed 
numbers  ? 

How  can  a  fraction  be  multiplied  by  a  fraction  ?  A  frac- 
tion by  an  integer? 

How  can  an  integer  be  multiplied  by  a  fraction?  By  a 
mixed  number? 

How  can  a  fraction  be  divided  by  a  fraction? 

How  do  you  reduce  a  complex  fraction  to  a  simple  frac- 
tion? 


COMPOUND   NUMBERS. 


180.  A  number  composed  of  only  one  kind  of  unit  is  a 
Simple  Number  ;  as,  5  pk.,  4  apples,  6. 

181.  A  Denomination  is  a  name  given  to  a  unit  of  measure 
or  of  weight. 

182.  A  number  composed  of  different  kinds  of  units  is 
a  Compound  Number ;  as,  3  bu.  2  pk.  1  qt. 

A   number   having   one   or   more  denominations  is  also 
called  a  Denominate  Number. 

183.  Reduction  is  the  process  of  changing  a  number  from 
one  denomination  to  another  without  changing  its  value. 

184.  Changing  to  a  lower  denomination  is  called  Reduction 
Descending;  as,  2  bu.  3  pk.  =  88  qt. 

185.  Changing  to  a  higher  denomination  is  called  Reduc- 
tion Ascending ;  as,  88  qt.  =  2  bu.  3  pk. 

186.  Linear  Measure  is  used  in  measuring  lines  or  distances. 

TABLE. 

12    inches  (in.)  =  1  foot,        ft. 

3    feet  =  1  yard,       yd. 

5|-  yards,  or  16^  feet  =  1  rod,  rd. 

40    rods  =  1  furlong,  fur. 

8    furlongs  =  1  mile,       mi. 

320    rods,  or  5280  feet  =  1  mile. 
1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63360  in. 
136 


136  COMPOUND    NUMBERS. 

187.  Surveyors'  Measure  is  used  in  measuring  land. 

TABLE. 

7.92  inches  =  1  link,     li. 
100  links     =  1  chain,  ch. 
80  chains  =  1  mile,    mi. 

Note.  — A  surveyors'  chain  is  4  rods  long,  and  contains  100  links. 
A  chain,  or  steel  measuring  tape,  100  feet  long,  is  sometimes  used 
by  engineers. 

188.  Square  Measure  is  used  in  measuring  surfaces. 


J  /  TABLE. 

y      "^     y    14:4:    square  inches    =  1  square  foot, 
M  aM  "^  /  "^  ^    square  feet         =  1  square  yard, 


sq.  ft. 

sq.  yd. 

304;  square  yards )        .  , 

o'roi  i    4.     r  =  1  square  rod,       sq.  rd. 

272J  square  feet     ^  ^  ?         ^ 


f    y#^<  ^vilGO    square  rods        =  1  acre,  A 

/       640    acres  =  1  square  mile,     sq.  mi. 

1  sq.  mi.  =  640  A.  =  102400  sq.  rd.  =  3097600  sq.  yd. 

189.  A  square  mile  of  land  is  called  a  Section. 

A  square  rod  is  sometimes  called  a  perch  (P.).     A  rood 
(R.)  is  40  sq.  rods. 

Note,  —  1  acre  =  43560  sq.   ft.     There   are   10   square   chains  in 
an  acre. 

Eoofing,  paving,  etc.,  are  often  estimated  by  the  Square, 
which  is  100  square  feet. 

190.  Cubic   Measure    is   used    in   measuring   volumes   or 
solids. 

TABLE. 

1728  cubic  inches  =  1  cubic  foot,  cu.  ft. 

27  cubic  feet  =  1  cubic  yard,  cu.  yd. 

16  cubic  feet  =  1  cord  foot,  cd.  ft. 

8  cord  feet,  or  128  cu.  ft.  =  1  cord,  cd. 
1  cu.  yd.  =  27  cu.  ft.  =  46656  cu.  in. 


TABLES.  137 

191.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE. 

4  gills  (gi.)  =  1  pint,       pt. 
2  pints         =  1  quart,     qt. 
4  quarts       =  1  gallon,    gal. 
1  gal.  =  4  qt.  =  8  pt.  =  32  gi. 

A  gallon  contains  231  cubic  inches. 

The  standard  barrel  is  31 J  gal.,  and  the  hogshead  63  gal. 

192.  Apothecaries'  Fluid  Measure  is  used  in  mixing  medi- 
cines in  liquid  form.  table 

60  minims  (ni)  —  1  fluid  dram,     f.  3. 
8  fluid  drams  =  1  fluid  ounce,    f.  S« 
16  fluid  ounces  =  1  pint  (0). 

193.  Dry  Measure  is  used  in  measuring  roots,  grain,  vege- 
tables, etc.  ^^3^^ 

2  pints    =  1  quart,         qt. 
8  quarts  =  1  peck,  pk. 

4  pecks   =  1  bushel,       bu. 
1  bu.  =  4  pk.  =  32  qt.  =  64  pt. 

The  bushel  contains  2150.42  cubic  inches. 

194.  Avoirdupois  Weight  is  used  in  weighing  all  common 
articles ;  as,  coal,  groceries,  hay,  etc. 

TABLE, 

16  ounces  =  1  pound,  lb. 

. ^„  ,  (1  hundred- weight,  cwt. 

100  pounds  =  ■{  ^  1  ., 

^  (or  cental,  ctl. 

20  cwt.,  or  2000  lb.  =  1  ton,  T. 

1  T.  =  20  cwt.  =  2000  lb.  =  32000  oz. 

The  Long  Ton  of  2240  pounds  is  used  at  the  U.  S.  Cus- 
tom-House  and  in  weighing  coal  at  the  mines. 


138  COMPOTTND    NUMBERS. 

The  ounce  is  considered  as  16  drams. 

The  Avoirdupois  pound  contains  7000  grains. 

A  hundred-weight  is  sometimes  called  a  Cental. 

195.  Troy  Weight  is  used  in  weighing  gold,  silver,  and 

jewels.  TABLE. 

24  grains  (gr.)      =  1  pennyweight,    pwt. 
20  pennyweights  =  1  ounce,  oz. 

12  ounces  =  1  pound,  lb. 

1  lb.  =  12  oz.  =  240  pwt.  =  5760  grains. 

196.  Apothecaries'  Weight  is  used  by  druggists  and  physi- 
cians in  weighing  medicines  that  are  not  liquid. 

TABLE. 

20  grains  (gr.)  =  1  scruple,  sc.  or  3. 
3  scruples  =  1  dram,  dr.  or  3. 
8  drams  =  1  ounce,        oz.  or  5 . 

12  ounces  =  1  pound,       lb.  or  lb. 

1  lb.  =  12  oz.  =  96  dr.  =  288  sc.  =  5760  gr. 

Dry  medicines  are  bought  and  sold  in  large  quantities 
by  Avoirdupois  weight. 

Comparison  of  Weights. 

1  lb.  Avoirdupois  =  7000  gr 

1  oz.  Avoirdupois  =  437^  gr. 

1  lb.  Troy  or  Apothecary  =  5760  gr. 
1  oz.  Troy  or  Apothecary  =  480    gr. 

197. 


Measure  of  Time. 

TABLE. 

60  seconds  (sec 

.)  =  1  minute,    min. 

60  minutes 

=  1  hour,         hr. 

24  hours 

=  1  day,          da. 

7  days 

=  1  week,       wk. 

365  days 

=  1  year,         yr. 

366  days 

=  1  leap  year. 

100  years 

=  1  century. 

TABLES.  139 

The  Civil  Day  begins  and  ends  at  midnight. 
■  The  exact  time  in  which  the  earth  makes  one  revolution 
of  the  sun  is  365  da.  5  hr.  48  min.  49.7  sec,  or  365J  days, 
nearly.  For  convenience  the  common  year  is  regarded  as 
365  days;  the  fraction  being  disregarded  until  it  amounts 
to  a  full  day,  which  is  in  four  years,  nearly.  Accordingly 
every  fourth  year  contains  366  days.  This  day  is  added 
to  the  shortest  month,  February,  and  the  year  in  which 
it  is  added  is  called  Leap  Year. 

But  365J  days  is  a  little  more  than  the  exact  year,  and 
we  have  added  a  little  too  much  when  we. have  added  1 
day  to  every  fourth  year,  therefore  only  every  fourth  cen- 
tennial year  is  considered  as  leap  year.  This  nearly  cor- 
rects the  excess,  so  that  the  error  is  less  than  1  day  in 
about  3600  years. 

Every  year  divisible  by  4,  and  every  centennial  year 
divisible  by  400,  is  a  leap  year. 

CIRCULAR  OR  ANGULAR  MEASURE. 

198.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line, 
every  point  of  which  is  equally  distant  from  the  centre. 

199.  The    bounding    line    of   a 
circle  is  the  Circumference. 

200.  Any  part  of  a  circumfer- 
ence is  an  Arc. 

^  to  J3  and  C  to  D  are  arcs  of  a 
circle. 

201.  A    straight    line    through 
the   centre   of   a  circle   terminat- 
ing  at  the   circumference   is   the  ^  circle. 
Diameter. 

202.  A  straight  line  from  the  centre  to  the  circumference 
is  the  radius ;  as,  E  to  D,  or  E  to  C. 


140  COMPOUND    NUMBERS. 

203.  The  circumference  of  every  circle  is  divided  into 
360  equal  parts  called  Degrees,  each  degree  into  60  parts 
called  Minutes,  and  each  minute  into  60  parts  called  Seconds. 

204.  An  Angle  is  the  difference  in  direction  between  two 
straight  lines.  The  point  of  meeting  is  the  Vertex.  The 
vertex  is  at  the  centre  of  a  circle,  and  the  angle  is  measured 
in  degrees  by  the  arc  between  its  sides.  Thus  BD  is  the 
measure  of  the  angle  BED. 


TABLE  OF  CIRCULAR  MEASURE. 

o 


60  seconds  (")  =  1  minute, 
60  minutes       =  1  degree, 
360  degrees        =  1  circumference,  Cir. 

Note. — An  arc  of  90  degrees  or  J  of  a  circumference  is  called 
a  quadrant.  A  degree  upon  a  great  circle  of  the  earth  is  69. 16  statute 
miles,  or  60  geographical  miles.     A  sign  is  an  arc  of  30  degrees. 

205.  Federal  Money  is  the  currency  of  the  United  States. 

TABLE. 

10  mills  =  1  cent,  ct.  10  dimes   =  1  dollar,  $. 

10  cents  =  1  dime,  d.  10  dollars  =  1  eagle,  E. 

The  gold  coins  of  the  United  States  are  the  double-eagle, 
eagle,  half-eagle,  quarter-eagle,  and  one-dollar  piece. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar, 
and  the  ten-cent  piece. 

The  five-cent  piece  is  nickel,  and  the  one-cent  piece 
bronze. 

206.  English  or  Sterling  Money. 

TABLE. 

4  farthings  =  1  penny,  d. 
12  pence  =  1  shilling,  s. 
20  shillings  =  1  pound,    £,  or  1  sovereign. 

The  coin  which  represents  the  Pound  Sterling  is  the 
Sovereign,  equal  in  value  to  $4.8665. 


REDUCTION.  141 

207.  Counting. 

TABLE. 

12  things  =  1  dozen,  doz. 

12  dozen  =  1  gross,  gr. 

12  gross    =  1  great  gross,   G.  gr. 

208.  Paper. 

TABLE. 

24  sheets  =  1  quire.  2  reams     =  1  bundle. 

20  quires  =  1  ream.  5  bundles  =  1  bale. 

REDUCTION  DESCENDING. 

209.  1.   Keduce  5  lb.  6  oz.  12  pwt.  6  gr.  to  grains, 

5  lb.  6  oz.  12  pwt.  6  gr. 

12 

—  Solution.  —  Since  there  are  12  oz.  m  1  lb.,  in  5  lb. 

^^  there  are  5  times  12  oz.  =  60  oz.  (add  6  oz.)  =  66  oz. 

6  Since  there  are  20  pwt.  in  1  oz.,  in  66  oz.  there 
QQ  oz.  are  m  times  20  pwt.  =  1320  pwt. "'(add  12  pwt.)  = 
20  1332  pwt. 

-1Q2Q  Since  there   are  24   gr.    in  1  pwt.,  in  1332  pwt. 

^  o  there  are  1332  times  24  gr.  =  31968  gr.  (add  6  gr. ) 

=  31974  gr. 

1332  pwt.  ^ 

24  E-educe  to  lower  denominations : 


5328 
2664 


^/2.   17  yd.  2  ft  9  in.  to  inches. 
^3.    46  rd.  4  yd.  2  ft.  to  feet. 
-4.    3  mi.  75  rd.  4  ft.  to  inches. 


31968 

6 

31974  ffr.  *^*    16  -^-  140  sq.  rd.  2Q>  sq.  yd.  to  square 

yards. 
»^6.   4  A.  15  sq.  rd.  4  sq.  ft.  to  square  inches. 
.^.    50  ch.  45  li.  to  links. 

^.    16  cu.  yd.  25  cu.  ft.  900  cu.  in.  to  cubic  inches. 
/9.   8  cd.  12  cu.  ft.  to  cubic  feet. 
-10.   15  gal.  3  qt.  1  pt.  to  pints. 


142  COMPOUND    NUMBERS. 

^11.  4  0.  6  f.  5  3  f.  3  25  ni  to  minims. 

^2.  7  bu.  3  pk.  5  qt.  1  pt.  to  pints. 

^13.  16|  bu.  to  quarts. 

'^  14.  25  lb.  5  oz.  16  pwt.  10  gr.  to  grains. 

^16.  2  T.  6  cwt.  10  lb.  14  oz.  to  ounces. 

\/l6.  16  tb.  5  §  4  3  2  3  11  gr.  to  grains. 

^7.  28°  14'  18"  to  seconds. 

^  18.  £18  15s.  Sd.  3  far.  to  farthings. 

"19.  27  da.  18  h.  49  min.  to  seconds. 

"20.  3  wk.  48  min.  52  sec.  to  seconds. 

-^21.  How  many  quires  in  a  bundle  of  paper? 

'^%2.    How  many  pints  in  a  cask  of  molasses  holding  84 
gallons  ? 

23.  How  many  articles  in  7  G.  gr.  5  gr.  ? 

24.  How  many  hours  in  10  years,  allowing  for  two  leap 
years  ? 

25.  How  many  inches  in  4  J  rods  ? 

26.  What  is  the  cost  of  10  miles  of  telephone  wire  at 
28  cents  a  pound,  if  a  pound  measures  75  ft.  ? 

27.  Find  the  number  of  square  inches  in  a  square  yard  ; 
square  feet  in  a  square  chain ;  cubic  inches  in  a  cubic  yard. 

28.  How  many  hours  in  the  month  of  February,  1896  ? 

29.  How  many  cubic  inches  in  5  gallons? 

30.  How  many  square  yards  in  4  sq.  miles  ? 

31.  How  many  square  feet  in  2^-  acres  ? 

32.  How  many  ounces  in  3  lb.  of  silver  ?     3  lb.  of  iron  ? 

'33.    If  I  buy  3  bu.  of  nuts  at  $  4  a  bushel,  and  sell  them 
at  5^  a  pint,  how  much  shall  I  lose  ? 
34.    How  many  ounces  in  a  long  ton  ? 


BEDUCTION.  143 

35.  At  $  12  a  ton,  what  will  f  of  a  ton  of  hay  cost  ? 

36.  In  1800  years  how  many  centuries  ? 

37.  If  you  can  count  sixty  a  minute,  how  long  will  it  take 
to  count  180000  ? 

38.  Through  how  many  degrees  does  the  hour-hand  of  a 
clock  pass  in  6  hours  ? 

39.  Through  how  many  degrees  does  the   minute-hand 
pass  in  6  hours  ? 

40.  What  will  3  reams  of  paper  cost  at  40^  a  quire  ? 

41.  Reduce  3  mi.  4  fur.  20  rd.  5  yd.  2  ft.  8  in.  to  inches. 

42.  Reduce  6  mi.  240  rd.  to  feet. 

43.  Reduce  3  A.  8  sq.  rd.  5  sq.  yd.  3  sq.  ft.  to  sq.  inches. 

44.  Reduce  16  cu.  yd.  9  cu.  ft.  3  cu.  in.  to  cu.  inches. 

45.  Reduce  58  cd.  to  cu.  feet. 

46.  Reduce  2  T.  3  ctl.  16  lb.  to  ounces. 

47.  Reduce  3  lb.  9  oz.  15  pwt.  12  gr.  to  grains. 

48.  Reduce  60  gal.  3  qt.  3  gi.  to  gills. 

49.  How  many  sheets  in  5  bales  of  paper? 

50.  Reduce  3  wk.  6  da.  5  hr.  to  minutes. 


REDUCTION  ASCENDING. 

210.    1.    Reduce  1306  gills  to  higher  denominations. 
1306  gi.  Solution.  — Since  in  1  pt.  there  are  4  gi., 


326  pt  4-  2  si.        '^"^  -^'^^^  »^*  t^6^^  ^^6  ^s  many  pints  as  4  gi. 
"TTTT;     7  is  contained  times  in  1306  gi. ,  or  326  pt.  and 

^  *  2  gi.  remainder. 


40  gal.  -t-3  qt.  since  in  1  qt.  there  are  2  pt.,  in  326  pt. 

40  gal.  3  qt.  2  gi.  there  are  as  many  quarts  as  2  pt.  is  con- 

tained times  in  326  pt. ,  or  163  qt. 
Since  in  1  gal.  there  are  4  qt. ,  in  163  qt.  there  are  as  many  gal- 
lons as  4   qt.    is  contained  times  in  163  qt.,  or  40  gal.,  and  3  qt. 
remainder. 

Therefore,  in  1306  gills  there  are  40  gal.  3  qt.  0  pt.  2  gi. 


144  COMPOUND    NUMBERS. 

2.    How  many  rods  in  334  yd.  ? 

5^  yd.  334  yd.  Solution.  — Since  in  1  rd. 

2"  2  there  are  5^  yd.,  in  334  yd. 


11  half  yd.  |  668  half  yd. ^^^^re  are  as  many  rods  as 

60  rd  +8  half  vd      ^^  Y^-  is  contained  times  in 
•^    *     334  yd.,  or  60  rd.,  and  4  yd. 
remainder.    334  yd.  -^  5|  yd.  =  668  iialf  yd.  -=-11  lialf  yd.    8  half  yd. 
=  4  yd.     334  yd.  =  60  rd.  4  yd. 

3.  Reduce  225932  in.  to  miles,  etc. 

4.  How  many  miles  and  rods  are  there  in  35640  ft.  ? 

5.  Reduce  19922544  sq.  in.  to  higher  denominations. 

6.  Reduce  762051  cu.  in.  to  cu.  yards,  etc. 

7.  How  many  cords  in  7424  cu.  ft.  ? 

8.  Reduce  69056  oz.  to  tons,  etc. 

9.  Reduce  21076  gr.  to  higher  denominations. 

10.  Reduce  1947  gi.  to  gallons,  etc. 

11.  How  many  bales  in  24000  sheets  of  paper? 

12.  Reduce  39180  min.  to  weeks,  etc. 

13.  Reduce  5762  far.  to  higher  denominations. 

14.  Reduce  84623"  to  higher  denominations. 

15.  Reduce  62341  ni  to  higher  denominations. 

16.  How  many  chains,  etc.,  in  13025  li.  ? 

17.  How  many  bushels,  etc.,  in  35842  pints? 

18.  How  many  pounds,  etc.  (Troy)  in  32563  gr.  ? 

19.  Reduce  39632  gr.  to  lb.,  etc.  (Apoth.). 

20.  How  many  tons,  etc.,  in  35682  lb.  ? 

21.  A  box  contains  75832  pens.     How  many  Gr.  gross, 
etc.,  in  the  box  ? 

22.  Change  1384  dry  pints  to  higher  denominations. 

23.  In  139843  sq.  in.  how  many  square  miles,  rods,  etc.  ? 

24.  How  many  cords  of  wood  in  3692  cu.  feet  ? 


REVIEW.  145 


REVIEW  PROBLEMS. 


211.    1.    Bought  2  gal.   8  oz.  of  fluid  extract  at  20^  an 
ounce,  and  sold  it  at  15^  an  ounce.     What  was  lost  ? 

2.  How  many  minims  are  there  in  10  fluid  ounces  (f.  S  ), 
7  fluid  drachms  (f.  3)  ? 

3.  Find  the  difference  in  value  b^ween  4  gal.  of  ammo- 
nia water  at  10  cents  a  pint  and  8  ounces  of  cinnamon  water 
at  5  cents  an  ounce. 

4.  The  pendulum  of  a  certain  clock  beats  seconds.  How 
many  times  will  it  tick  in  1  day,  9  hours,  25  minutes  ? 

5.  How  many  degrees  in  3492.58  statute  miles,  measured 
on  the  equator,  a  degree  being  equal  to  69.16  statute  miles  ? 

6.  How  many  degrees  of  longitude  will  a  steamship  pass 
through,  sailing  due  west  on  the  equator,  at  the  rate  of  15 
knots  an  hour  for  5  days  ? 

Note.  —  A  knot  =  1  geographic  mile  or  minute. 

7.  Find  cost  of  each  of  the  following: 

(a)  5  gallons,  3  qt.  1  pt.  of  molasses  at  20^  a  gallon ; 

(b)  10  acres,  50  sq.  rd.  of  land  at  $  50  an  A. 

8.  What  will  it  cost  to  build  112  rd.  3  yd.  of  fence  at 
48^  a  yard  ? 

9.  If  a  man  steps  2^  ft.  at  each  step,  how  many  mile's 
will  he  travel  in  stepping  4820  times  ? 

10.  If  17  ft.  is  f  of  the  height  of  a  tree,  how  high  is  the 
tree? 

11.  Eeduce  6|  mi.  317  rd.  4  yd.  2  ft.  to  feet. 

12.  Change  16571  ft.  to  miles. 

13.  At  $3.20  a  bu.  how  many  quarts  of  nuts  can  be 
bought  for  $  4.80  ? 

14.  How  many  pint  bottles  of  camphor  may  be  filled 
from  96  fluid  ounces  (f .  3  )  ? 


146  COMPOUND   NFMBEES. 

15.  Find  the  cost  of  the  following:  4  oz.  iodine  at  10^, 
8  oz.  spts.  camphor  at  5^,  10  oz.  aqua  ammonia  at  10^,  14 
oz.  cinnamon  water  at  5^. 

16.  Reduce  4  bu.  3  pk.  7  qt.  1  pt.  to  pints. 

17.  How  many  quart  boxes  will  hold  2  bu.  3  pk.  5  qt.  of 
berries  ? 

18.  If  4  bu.  of  bef ries  are  bought  for  $  .70  per  bushel 
and  sold  for  $  .05  per  quart,  what  is  the  gain  ox  loss  ? 

19.  Both  sides  of  a  railroad  track  are  fenced  with  wire 
for  40  yards.     What  is  the  cost  of  the  fence  at  4^  a  foot.  ? 

20.  What  will  8  lb.  6  oz.  of  sugar  cost  at  8^  a  pound  ? 

21.  When  pens  are  bought  at  75^  a  gross,  and  sold  at  2 
for  3^,  what  is  the  gain  ? 

22.  If  a  man  can  walk  10  miles  in  2  hours,  how  far  can 
he  walk  in  6  hours  ?     In  30  minutes  ?     In  50  minutes  ? 

23.  What  will  ^  bu.  berries  bring  at  8^  a  quart  ? 

24.  A  silver  chain  weighs  18  pwt.  What  is  its  value, 
when  silver  is  worth  $  .65  an  ounce  ? 

25.  What  will  24  qt.  of  milk  cost  at  20^  a  gallon  ? 

26.  If  I  buy  peanuts  at  5^  a  quart,  and  retail  them  so  as 
to  gain  $  6.40  on  4  bushels,  what  do  I  sell  them  for  ? 

'27.    At  4  pens  for  3  cents  what  will  1  great  gross  cost  ? 

28.  How  many  table-forks,  each  weighing  2J  oz.,  can  be 
made  from  4  lb.  4  oz.  10  pwt.  of  silver  ? 

29.  In  f  of  a  gallon  how  many  pints  ? 

30.  How  many  rods  of  fence  will  enclose  a  mile  square 
of  land  ? 

31.  What  is  the  cost  of  1  yd.  and  27  inches  of  fringe  at 
60  cents  a  yard  ? 

32.  How  many  rods  of  fence  are  required  to  enclose  a  lot 
that  is  20  rods  wide  and  three  times  as  long  ? 


BEDUCTION.  147 

33.  Required  the  distance  around  a  room  that  is  13  feet 
long  and  15  feet  wide. 

34.  A  shoe-box  is  4  in.  deep,  6  in.  wide,  and  12  in.  long. 
How  much  twine  will  it  take  to  wind  twice  around  the  box 
each  way  to  hold  on  the  cover,  allowing  6  inches  for  tying.'/ 

35.  I  have  a  lawn  that  is  30  ft.  by  70  ft.,  and  wish  to  lay 
a  board  walk  around  it  that  is  3  ft.  6  in.  in  width.  What 
is  the  distance  around  the  walk,  outside  measurement  ? 

212.  A  Denominate  Fraction  is  a  fraction  having  a  de- 
nomination. 

213.  To  reduce  denominate  fractions  to  Integers  of  Lower 
Denominations. 

1.  Reduce  f  of  a  mile  to  rods,  yards,  feet,  etc. 

SoLDTioK.  —  f  of  320  rd.  =  i-«^  rd.  =  228f  rd. 
I  of  -\i  yd.  =  ff  yd.  =  3f  yd. 
f  of  3  ft.  =  Of  ft. 
f  of  12  in.  =  -3^  in.  =  5}  in. 
f  of  a  mile  =  228  rd.  3  yd.  0  ft.  5^  in. 

Note.  —  The  same  process  applies  to  denominate  decimals. 

2.  Reduce  .87  bu.  to  pecks,  quarts,  etc. 

.87  bu.  .87  of  4  pk.  =  3.48  pk. 


4 

3.48 

g 

.48  of  8  qt.  =  3.84  qt. 

3.84 
2 

'.84  of  2pt.  =1.68pt. 

1.68 

.87  bu.  z=  3  pk.  3  qt.  1.68  pt. 

Rule.  —  Change  the  given  fraction  (or  decimal)  to  the  next 
lower  denomination.  Treat  the  fractional  (or  decimal) 
part  of  the  product  in  the  same  way,  and  so  proceed  to 
the  required  denominatioii. 


148  COMPOUND   NUMBERS. 

Eeduce  to  integers  of  lower  denominations. 

3.  I  of  a  mile.  9.  .375  of  a  month. 

4.  f  of  an  acre.  10.  .3125  of  a  gallon. 

5.  I"  of  a  pound  (Troy).  11.  .4267  of  an  acre. 

6.  f  of  a  ton.  12.  .2364  of  a  ton. 

7.  f  of  a  gallon.  13.  .363  of  a  sign. 

8.  I  of  a  mile.  14.  .51625  of  a  mile. 

15.  Reduce  ^|  mi.  to  lower  denominations. 

16.  Change  f  of  a  year  to  months  and  days. 

17.  In  -^  gal.  how  many  qt.  and  pt.  ? 

18.  Reduce  -^^  lb.  to  oz.  and  dr. 

19.  -f^  acre  are  equal  to  how  many  sq.  rods,  etc.  ? 

20.  Reduce  f^  bu.  to  integers  of  lower  denominations. 

21.  What  is  the  value  of  ^  of  ^  of  a  hhd.  in  integers  of 
lower  denominations  ? 

22.  What  is  the  value  of  ^7^  of  an  acre  in  integers  of 
lower  denominations  ? 

23.  Reduce  £  J  to  integers  of  lower  denominations. 

24.  W^hat  is  the  value  of  ^  of  1^  of  a  mile  ? 

214.   To  reduce  denominate  numbers  to  Fractions  of  Higher 
Denominations. 

1.  Reduce  2  qt.  1  pt.  2  gi.  to  the  fraction  of  a  gallon. 

Solution.  —  2  gi.  -^  4  =  |  pt.  =  ^  pt. 

Upt.  =  f  pt.  .  f  pt. -2  =  |qt. 
2f  qt.  =  J^  qt.  -^  4  =  \\  gal.      Ans. 

2.  Reduce  2  qt.  1  pt.  2  gi.  to  the  decimal  of  a  gallon. 


BEDUCTION.  149 


gi.  Rule.  —  Change  the  number  of  the  lowest 


1.5  pt. denomination  to  a  fraction  (or  deci- 


2.75  qt.  mal)    of  the   next   higher,   write    this 


6875  eal  fraction   (or  decimal)  as  a  part  of 

the  number  of  that  higher  denominor 
Hon,  and  reduce  this  number  in  like  manner,  and  so 
proceed  to  the  required  denomination. 

3.  Reduce  213  rd.  1  yd.  2  ft.  6  in.  to  a  fraction  of  a 
mile. 

4.  What  fraction  of  an  acre  is  3  E,.  13  sq.  rd.  10  sq.  yd. 
108  sq.  in.  ? 

5.  What  part  of  a  year  is  273  da.  18  hr.  ? 

6.  Reduce  to  a  fraction  of  a  pound  8  oz.  11  pwt.  10|-  gr. 

7.  What  part  of  a  ton  is  857  lb.  2f  oz.  ? 

8.  Change  3  fur.  19  rd.  5  yd.  1  ft.  4.7328  in.  to  the  deci- 
mal of  a  mile. 

9.  Reduce  1  da.  14  hr.  24  min.  to  the  decimal  of  a 
month. 

10.  What  decimal  of  a  gallon  is  1  qt.  2  gi.  ? 

11.  Reduce  68  sq.  rd.  8  sq.  yd.  2  sq.  ft.  7.488  sq.  in.  to 
the  decimal  of  an  acre. 

12.  What  decimal  of  a  pound  Troy  is  6  oz.  3  pwt.  21.6 
gr.? 

13.  Reduce  131  da.  18  hr.  21  min.  36  sec.  to  the  decimal 
of  a  year. 

14.  Reduce  2  qt.  If  gi.  to  the  fraction  of  a  gallon. 

15.  What  fraction  of  a  mile  is  71  rd.  1  ft.  10  in.  ? 

16.  Reduce  12  da.  34  min.  17^^  sec.  to  the  fraction  of 
a  month. 

17.  What  decimal  of  a  ton  is  4  cwt.  72  lb.  128  oz.  ? 

18.  Reduce  48  cu.  ft.  1636.7616  cu.  in.  to  the  decimal  of 
a  cord. 


150  COMPOUND   NUMBERS. 

19.  What  decimal  of  a  circle  is  10°  53'  24"  ? 

20.  Reduce  4  fur.  5  rd.  1  yd.  3.6  in.  to  the  decimal  of  a 
mile. 

21.  Reduce  6  pwt.  to  a  fraction  of  a  pound. 

22.  3  qt.  1  pt.  2  gi.  are  what  part  of  a  gallon  ? 

23.  Change  6  rd.  4  yd.  1  ft.  to  the  fraction  of  a  mile. 

24.  What  part  of  a  cord  of  wood  are  8  cu.  ft.  ? 

25.  Reduce  5  gross  7  doz.  to  the  fraction  of  a  great  gross. 

To  find  what  part  one  denominate  number  is  of  another. 

I.  What  part  of  2  gal.  1  qt.  1  pt.  is  3  qt.  1  pt.  1  gi.  ? 

3  qt.  1  pt.  1  gi.  =  29  gi. 
2  gal.  1  qt.  1  pt.  =  76  gi. 
The  question  now  is,  29  gi.  is  what  part  of  76  gi.  ? 
29  gi.  is  ^  of  76  gi.     Ans. 
Note.  —  To  find  the  decimal  part,  divide  numerator  by  denominator. 

2.  What  part  of  5  lb.  9  oz.  3  pwt.  is  2  lb.  8  oz.  6  pwt. 
10  gr.  ? 

3.  What  part  of  3  mi.  24  rd.  5  yd.  is  2  mi.  34  rd.  4  yd.  ? 

4.  What  part  of  3  da.  5  hr.  22  min.   is  1  da.  10  hr. 
3  min.  12  sec.  ? 

5.  What  decimal  of  3  gal.  2  qt.  1  pt.  is  2  gal.  2  qt. 
2pt.? 

6.  What  decimal  of  4  T.  5  cwt.  10  lb.  is  2  T.  6  cwt. 
13  lb.  ? 

7.  What  part  of  a  rod  is  4  yd.  2  ft.  7  in.  ?' 

8.  What  part  of  6  rods  is  f  of  7  feet  ? 

9.  What  part  of  3|  mi.  is  160  rd.  5  yd.  ? 
10.    ^  pint  is  what  part  of  a  bushel  ? 

II.  What  decimal  of  8  bu.  3  pk.  4  qt.  is  4  bu.  1  pk.  5  qt.  ? 


ADDITION.  151 


ADDITION   OF  COMPOUND   NUMBERS. 

215.     1.    Add  14  lb.   5  oz.  17  pwt.  12  gi.,  18  lb.  10  oz. 
14  gr.,  6  lb.  4  oz.  8  pwt.  16  gr. 


lb. 

oz. 

pwt. 

gr- 

14 

5 

17 

12 

18 

10 

0 

14 

6 

4 

8 

16 

39 

8 

6 

18 

Solution.  —  The  sum  of  the  grains  =  42  gr. 
=  1  pwt.  18  gr.  We  place  the  18  gr.  under  the 
column  of  grains,  and  add  the  1  pwt.  to  the  col- 
umn of  pennyweights.  Add  the  other  columns 
in  like  manner. 


rd.       yd.      ft.  rd.         ft, 

2.  17       4       1  3.  6       12 


12 

4 

2 

4 

14 

11 

6 

5 

n 

17 

15 

9 

8 

3 

2 

6 

12 

8 

46 

H 

n 

36 

5^ 

10 

u= 

.^yd. 

6  =  ^ft. 

46       2       0  m 


tons.  cwt.   lb.   oz. 

4. 

14   13  Q>^      15 

13   17   88   11 

46   16  86   13 

14   15  57   6 

11   17   85   15 

deg.  min.   sec. 

6. 

29  59  59 

15   45   42 

18   11   40 

13  19   17 

sq.  yd.     sq.  ft.      sq.  in. 


6. 


45    8 

113 

45    3 

112 

75    8 

139 

49    0 

115 

589    8 

90 

yr.    da.   hr. 

min. 

see. 

18  345   13 

37 

15 

87  169  12 

16 

28 

316  144  20 

53 

18 

13  360  21 

57 

15 

152  COMPOUND   NUMBERS. 

bu.      pk.      qt.      pt.  cd.        cd.  ft.       ctu  ft 

8. 


40 

2 

6 

1 

9. 

5 

7 

0 

89 

1 

3 

0 

2 

2 

12 

75 

2 

1 

1 

0 

6 

15 

69 

2 

3 

0 

'^1 

0 

0 

49 

1 
3 

2 
1 

1 
1 

3 

0 

2 

65 

10.  Find  the  sum  of  3  T.  15  ewt.  25  lb.  9  oz.,  4  T.  17 
cwt.  30  lb.  10  oz.,  6  T.  18  cwt.  15  lb.  12  oz.,  2  T.  12  cwt. 
20  lb.  16  oz. 

11.  Find  the  sum  of  7  hr.  30  min.  45  sec,  12  hr.  25  min.. 
30  sec,  20  hr.  15  min.  33  sec,  10  hr.  27  min.  46  sec 

12.  Add  10  mi.  101  rd.  3  yd.  2  ft.  11  in.,  16  mi.  4  yd. 
6  in.,  3  mi.  560  rd.  3  ft.,  175  rd.  4  ft.  7  in. 

13.  Add  3  A.  50  sq.  rd.  25  sq  yd.  10  sq.  ft.  102  sq.  in., 
5  A.  110  sq.  rd.  30  sq.  yd.  8  sq.  ft.  34  sq.  in.,  6  A.  75  sq. 
rd.  14  sq.  yd.  7  sq.  ft.  82  sq.  in.,  7  A.  215  sq.  rd.  17  sq.  yd. 
16  sq.  ft.  53  sq.  in. 

14.  Find  the  sum  of  18  cd.  6  cd.  ft.  12  cu.  ft.,  19  cd.  4  cd. 
ft.  6  cu.  ft.,  24  cd.  2  cd.  ft.  1  cu   ft. 

15.  Find  the  sum  of  18  T.  18  lb.  12  oz.,  16  cwt.  21  lb., 
14  cwt.  75  lb.  10  oz. 

16.  What  is  the  entire  length  of  a  railway  consisting  of 
5  different  lines  measuring  respectively  160  mi.  185  rd.  2 
yd.,  97  mi.  63  rd.  4  yd.,  126  mi.  272  rd.  3  yd.,  67  mi.  199  rd. 
5  yd.,  and  48  mi.  266  rd.  5  yd.  ? 

17.  A  merchant  sold  48  gal.  3  qt.  1  pt.  of  coal  oil  and 
had  15  gal.  1  qt.  1  pt.  left.     What  quantity  had  he  at  first  ? 

18.  A  starts  from  a  point  in  Lat.  21°  25'  35"  N.  and 
travels  north  24°  36'  45".     At  what  latitude  does  he  arrive  ? 

19.  Find  the  difference  in  longitude  between  a  point 
46°  15'  30"  E.  and  a  point  21°  18'  16"  W. 


SUBTRACTION.  153 

Note,  —  When  one  place  is  in  east  and  the  other  in  west  longi- 
tude, their  difference  in  longitude  is  the  sum  of  their  longitudes. 

20.  Charles  walks  5  mi.  15  rd.  2  ft.  north  of  the  school- 
house,  and  James  6  mi.  28  rd.  5  yd.  south.  How  far  are 
they  apart? 

21.  Find  the  sum  of  f  mi.  35  rd.  4|  rd. 

Note.  —  Reduce  each  to  integers  of  lower  denominations,  then  add. 

22.  Add  f  bu.  17|  pk.  4f  pt.,  6  bu.  3|  pk.  2  qt.,  J  bu. 
J  pk.  5  qt. 

23.  What  is  the  sum  of  f  T.  f  cwt.  and  f  lb.  ? 

SUBTRACTION   OF   COMPOUND   NUMBERS. 

lb.     oz.    pwt.     gr.  Solution.  —  15   gr.  -  12 

216.    1.    From    6       2     14     15      S^- = '^  ^'-    ^«  ^^  ^^^"^^ 

Take    4     10     18     12      ^^^^,  !f  T''  'T  l^  ^"\ 
we  take  1  oz.,  which  equals 

1        3     16        3      20    pwt.,    and    add    to    the 

14  pwt.  =  34  pwt.  ;  34  pwt. 

—  18  pwt.  =16  pwt.     We  have  taken  1  oz.  from  the  2  oz.,  leaving 

1  oz.  ;  but  as  we  cannot  take  10  oz.  from  1  oz.,  we  take  1  lb.  =  12  oz., 

and  add  it  to  1  oz.  =  13  oz.,  from  which  take  10  oz.  =  3  oz.     Since  we 

took  1  of  the  6  lb.,  we  have  5  left ;  from  which  take  4  lb.  =  1  lb. 


2. 

3. 

A.    sq.  rd.  i 

sq.  ft. 

hr.  min.   sec. 

From 

10     50 

7 

5   54   30 

Take 

4    106 

5 

1   17   50 

4. 

'> 

5. 

gal.  qt.  pt.  gi. 

A. 

R.  sq.  rd.  sq.  yd. 

sq.  ft 

From 

39  2  2  1 

5 

1   39    15 

7 

Take 

16  2  3  3 
6. 

2 

2   26  •   21 
7. 

8 

da. 

hr.  miu.  sec. 

T.  cwt.  lb.   oz. 

200 

17  54  36 

20  15  75  10 

135 

20  24  48 

5  16  25  12 

154  COMPOUND   NUMBERS. 

8.  From  260  mi.  take  23  mi.  7  fur.  25  rd.  5  yd.  2  ft. 
10  in. 

9.  A  man  having  i  an  acre  of  ground  sold  25  sq.  rd.  11 
sq.  yd.  8  sq.  ft.  to  one  man,  and  50  sq.  rd.  9  sq.  yd.  4  sq.  ft. 
to  another.     How  much  land  had  he  left  ? 

10.  From  12  cwt.  subtract  9  cwt.  14  lb.  12  oz. 

11.  From  a  hogshead  of  molasses  25  gal.  3  qt.  2  pt.  were 
drawn  at  one  time,  and  at  another  time  10  gal.  1  pt.  How 
many  gallons  remained  ?  . 

12.  From  2  bu.  3  pk'.,  1  bu.  2  pk.  6  qt.  were  sold.  How 
much  remained  ? 

13.  An  apothecary  bought  2  lb.  of  quinine,  and  sold  1  lb 
3  oz.  5  dr.  2  sc.  11  gr.     How  much  had  he  left  ? 

14.  What  is  the  difference  in  longitude  between.  New 
York  74°  0'  3"  W.  and  San  Francisco  122°  25'  40"  W.  ? 

Note. — When  both  places  are  in  east  or  in  west  longitude,  their 
difference  of  longitude  is  found  by  subtraction. 

15.  Charles  walks  22  mi.  4  rd.  2  yd.  south  of  the  school, 
and  Henry  16  mi.  160  rd.  3  yd.  in  the  same  direction.  How 
far  are  they  apart  ? 

16.  From  J  of  a  mile  take  16\  rd. 

Note.  —  Change  both  to  integers  of  lower  denominations,  then 
subtract. 

17.  Rome  is  in  longitude  12°  28' 40"  E.,  and  Paris  in 
longitude  2°  20'  14"  E.  What  is  their  difference  in  longi- 
tude ? 

18.  From  |  of  a  pound  Avoir,  take  3f  oz. 

19.  From  16|  bu.  take  71  pk. 

20.  Take  .325  T.  from  6.54|  cwt. 

21.  Take  .7  of  a  rod  from  4  yd.  2  ft.  8  in. 

22.  From  22  da.  16  hr.  20  min.  take  2^  weeks. 


SUBTRACTION.  155 

DIFFERENCE  BETWEEN  DATES. 
217.    1.    Find  the  time  from  Jan.  25, 1842,  to  July  4, 1896. 

1896  7  4  Solution.  —  It  is  customary  to  con- 

.  „  . ,.  .  „^  sicler  80  days  to  a  month.     July  4, 1896,  is 

^^  1  wO  ^^^  1896th  yr.  7th  mo.  4th  da.,  and  Jan. 

54  yr.  5  mo.    9  da.      25,  1842,  is  the  1842d  yr.  1st  mo.  25th  da. 

Subtract,  taking  30  da.  for  a  month. 

2.  What  is  the  exact  number  of  days  between  Dec.  16, 
1895,  and  March  12,  1896  ? 

j-/c«^.    Ltj  Solution. — Do  not  count   the  first  day 

Jan.   31  mentioned.    There  are  15  days  in  December, 

Feb.   29  after  the  16th.     January  has  31  days,  Feb- 

Mar.  12  ruary  29  (leap  year),  and  12  days  in  March ; 

~  -               making  87  days.     Ans.  ■ 

3.  How  much  time  elapsed  from  the  landing  of  the  Pil- 
grims, Dec.  11,  1620,  to  the  Declaration  of  Independence, 
July  4,  1776  ? 

4.  How  much  time  elapsed  from  the  beginning  of  the 
Civil  War,  April  14,  1861,  to  the  close  of  the  war,  April  9, 
1865? 

5.  Washington  was  born  Feb.  22,  1732,  and  died  Dec. 
14,  1799.     How  long  did  he  live  ? 

6.  Washington  was  first  inaugurated  April  30,  1789. 
How  long  ago  was  his  inauguration  ? 

7.  How  much  time  will  have  elapsed  since  Columbus 
discovered  America,  Oct.  12,  1492,  to  your  next  birthday? 

8.  Mr.  Smith  gave  a  note  dated  Feb.  25,  1896,  and  paid 
it  July  12,  1896.  Find  the  exact  number  of  days  between 
its  date  and  time  of  payment. 

9.  A  carpenter  earning  $  2.50  per  day,  commenced 
Wednesday  morning,  April  1,  1896,  and  continued  work- 
ing every  week  day  until  June  6.     How  much  did  he  earn  ? 


156  COMPOUND   NUMBERS. 

10.  Fred  was  born  Dec.  20,  1875  ;  how  old  is  he  now  ? 

11.  How  much  time  has  elapsed  since  George  Washing- 
ton was  15J  years  old  ? 

12.  General  Grant  was  born  April  27,  1822.     How  old 
would  he  be  if  he  were  alive  to-day  ? 

13.  How  long  since  Lee  surrendered  to  General  Grant? 

14.  Find  the   exact   number  of   days   between  Jan.  10, 
1896,  and  May  5,  1896. 

15.  When  can  a  boy  who  was  born  May  5,  1882,  cele- 
brate his  25th  birthday  ? 

16.  John  goes  to  bed  at  9.15  p.m.  and  gets  up  at  7.10  a.m. 
How  many  minutes  does  he  spend  in  bed  ? 

MULTIPLICATION  OF  COMPOUND  NUMBERS. 
218.    1.    Multiply  4  yd.  2  ft.  8  in.  by  8. 

Solution.  —  8  times  8  in.  =  64  in.  =  5  ft.  4  in. 

^  ■        '         '         Place  the  4  in.  under  the  inches'  column,  and  reserve 

the  5  ft.  to  be  added  to  the  product  of  2  ft.  by  8, 

5  which  equals  16  ft.  (add  5  ft.)=  21  ft.     21  ft.  --  3 

39       0       4  =7   yd.,  with  no  remainder.      Add  7  yd.  to  the 

product  of  4  yd.  by  8  =  32  yd.  (add  7  yd.)  =  .39  yd. 

2.  gal.    qt.    pt.    gi.  bu.    pk.    qt.    pt. 

31     3    2    3  12    3     2      1 

5  8 

/-^/  /      /    A  

3.  If  a  man  travel  at  the  rate  of  60  mi.  240  rd.  16  ft.  in 
one  day,  how  far  will  he  travel  in  ten  days  ? 

4.  A  man  owns  6  farms,  each  containing  75  A.  49  sq.  rd. 
25  sq.  yd.  of  land.     How  much  land  in  all  the  farms  ? 

5.  If  6  loads  of  hay  weigh  6  T.  18  cwt.  75  lb.,  how 
much  will  48  loads  weigh  ? 

Note.  —  48  loads  will  weigh  8  times  as  much  as  6  loads. 


MULTIPLICATION.  157 

6.  If  12  spoons  weigh  3  lb.  8  oz.  15  pwt.,  how  much 
will  one  gross  of  spoons  weigh? 

7.  How  much  oil  will  7  barrels  hold  if  each  barrel  con- 
tains 35  gal.  2  qt.  ? 

8.  What  is  the  value  at  ^  4  per  cord  of  10  piles  of  wood, 
each  containing  5  cd.  5  cd.  ft.  12  cu.  ft.  ? 

9.  What  is  the  weight  of  15  packages,  each  weighing 
1  lb.  4  oz.  (Avoir.)  ? 

10.  If  a  bicyclist  travels  75  mi.  140  rd.  in  one  day,  how 
far  can  he  travel  in  ten  days  ? 

11.  In  a  watch-chain  there  are  2  oz.  12  pwt.  15  gr.  of 
gold.     How  much  gold  is  required  for  25  such  chains  ? 

12.  A  farmer  has  six  bins,  each  containing  60  bu.  2  pk. 
of  wheat.     How  much  wheat  has  he  ? 

13.  If  a  train  is  run  for  8  hours  at  the  average  rate  of 
50  mi.  30  rd.  10  ft.  per  hour,  how  great  is  the  distance 
covered  ? 

14.  It  takes  John  Smith  5  hr.  20  min.  11  sec.  to  plough 
one  acre  of  ground.  At  the  same  rate,  how  long  will  it 
take  him  to  plough  4  acres  ? 

15.  4  gal.  3  qt.  1  pt.  X  11  =  ? 

16.  2  A.  40  sq.  rd.  16  sq.  yd.  x  20  =  ? 

DIVISION  OF   COMPOUND  NUMBERS. 

219.    1.    Divide  16  lb.  9  oz.  17  pwt.  8  gr.  by  10. 
Solution.  — J^  of  16  lb.  =  1,  and  6  lb.  remaining.     6  lb.  =  72  oz. 

IK     ^r,   «^f     „.         72  oz.  +  9  oz.  =  81  oz.     ^  of  81  oz.  =  8  oz., 
lb.    oz.  pwt.    gr.  1"  ' 

10 "116     9     17     8  ^'^^^  '^  °^*  ^^maining,  =  20  pwt.,  to  which 

^—^ — •     add  17  pwt.,  =  37  pwt.     J^  of  37  pwt.  =  3 

TIF     pwt.,    with    7    pwt.    remaining,  =  168    gr., 

to  which  add  8  gr.  ;  and  taking  ^-^  of  the  sum,  we  have  17j%  gr, 

"\Mien  the  divisor  is  large,  employ  long  division. 


158  COMPOUND   NUMBERS. 

2.    Find  ^  of  42  rd.  4  yd.  2  ft.  8  in. 


Solution.  —  ^^  of  42  rd.  =  1  rd.  ;  re- 
mainder, 7  rd.  =  38^  yd. ;  add  4  yd.  = 
421  yd.  ^1^  of  42-1  yd.  =  1  yd. ;  remainder, 
7-1  yd.,  =  22ift. ;  add  2  ft.  =  24^  ft.  j\  of 
24^  ft.  =  0  ft.  24^  ft.  =  294  in. ;  add  8 
m.  =  302  in.     ^^  of  302  in.  =  8|f  in. 

Note.  —  When  both  dividend  and  di- 
visor are  compound,  reduce  them  to  tbe 
same  denomination,  and  divide.  The 
quotient  will  be  abstract. 

3.  Divide  169  bu.  3  pk.  5  qt. 
by  7. 

4.  If  a  man  travelled  607   mi.. 
169   rd.    11  ft.   6   in.   in  10  days, 
what  average  distance  did  he  travel 
in  1  day  ? 

5.  If  one  gross  of  spools  weighs 
44  lb.  9  oz.,  how  much  will  one 
dozen  weigh  ? 

6.  If  one  bottle  holds  1  pt.  3  gi., 
how  many  dozen  bottles  will  be 
required  to  hold  65  gal.  2  qt. 
1  pt.  ? 

7.  A  man  has  451  A.  138  sq.  rd.  29  sq.  yd.  of  land,  which 
he  wishes  to  divide  equally  among  his  six  children.  How 
much  land  will  each  child  receive  ? 

8.  If  12  persons  share  equally  in  the  contents  of  a  bin 
containing  20  bu.  2  pk.  4  qt.  of  apples,  what  is  the  share 
of  each  ? 

,9.    If  the  entire  area  of  24  equal  fields  is  242  A.  20  sq. 
rd.  15  sq.  yd.,  what  is  the  size  of  each  field  ? 


rd.  yd.  ft. 

in. 

35)42  4  2 

8(1  rd. 

35 

7 

^ 

H 

35 

38i 

+  4 

35)42J-yd.  (1 

yd. 

35 

H 

3 

221  ft. 

+  2 

35)241.  ft.  (Oft. 

12 

294 

+  8 

35)302(8^2  ii, 

I. 

280 

22 

1  rd.  1  yd. 

m  in. 

Ans. 

DIVISION.  159 

10.  A  man  walked  50  mi.  71  rd.  2  yd.  in  15  hours. 
What  was  his  rate  per  hour? 

11.  If  it  takes  a  man  12  hr.  35  min.  15  sec.  to  walk  45 
miles,  what  is  the  average  time  taken  for  each  mile? 
(Divide  by  the  factors  of  45.) 

12.  AVhen  .f  12  will  buy  11  gal.  2  qt.  1  pt.  of  maple 
syrup,  how  much  will  $1  buy? 

13.  A  man  travelled  100  miles  in  9  hours.  What  was 
the  average  rate  per  hour? 

14.  If  a  horse  eats  12  qt.  of  oats  per  day,  how  long  will 
10  bu.  1  pk.  4  qt.  last  him  ? 

15.  If  a  package  weighs  4  cwt.  15  lb.,  how  many  such 
packages  will  it  take  to  weigh  3  T.  2  cwt.  25  lb.  ? 

16.  A  man  had  5  acres  of  land  which  he  divided  into  12 
equal  parts.     How  much  land  did  each  part  contain  ? 

17.  Divide  102  T.  15  cwt.  27  lb.  9  oz.  by  8. 

18.  Divide  16  bu.  3  pk.  6  qt.  by  2  bu.  1  pk. 

19.  I  have  84  lb.  14  oz.  of  salt  which  I  wish  to  put  into 
packages  of  2  lb.  6  oz.  each.  How  many  packages  will 
there  be? 

20.  If  a  horse  eats  1  pk.  2  qt.  of  oats  a  day,  how  many 
days  will  16  bu.  3  pk.  6  qt.  last  him  ? 

21.  How  many  sacks,  each  containing  2  bu.  3  pk.  2  qt., 
will  be  needed  to  hold  165  bu.  2  pk.  of  meal  ? 

22.  16  cwt.  75  lb.  9  oz.  of  butter  are  to  be  jjut  into  jars 
each  containing  9|  lb.     How  many  jars  will  be  needed  ? 

To  multiply  or  divide  a  compound  number  by  a  fraction. 

Note.  —  To  multiply  by  a  fraction,  multiply  by  the  numerator, 
and  divide  the  product  by  the  denominator. 

To  divide  by  a  fraction,  multiply  by  the  denominator,  and  divide 
the  product  by  the  numerator. 


160  COMPOUND   NUMBERS. 

23.  How  much  is  f  of  16  hr.  17  min.  14  sec.  ? 

24.  How  much  is  J  of  30  S  ^  3  1  3  8  gr. 

25.  Divide  120  cd.  50  cd.  ft.  34  cu.  in.  by  f . 

26.  How  many  times  is  ^|^  contained  in  840  T.  15  cwt. 
98  lb.  3  oz.  ?        . 

27.  A  man  sold  4  bu.  3  pk.  2  qt.  of  potatoes,  which  was 
■J  of  what  he  raised.     How  much  did  he  raise  ? 

28.  A  butcher  sells  120  tons,  9  cwt.  75  lb.  of  beef  every 
month.     How  much  does  he  sell  in  |  of  a  month  ? 

29.  If  6  bottles  hold  5  gal.  2  qt.  of  milk,  how  much  milk 
will  3  such  bottles  hold  ? 

30.  A  field  contains  10  acres  12  sq.  rd.  of  land,  which 
is  f  the  size  of  the  whole  farm.  Find  the  size  of  the 
farm.  • 

31.  'A  railroad  track  extends  144  miles,  40  rd.  3  yd. 
How  far  has  a  train  of  cars  gone  which  has  travelled  -^^  of 
this  distance  ? 

32.  If  a  pipe  discharges  25  gal.  3  qt.  1  pt.  of  water  in 
1  hr.,  how  much  will  it  discharge  in  5j  hr.,  if  the  water 
flows  with  the  same  velocity  ? 

Note.  —  When  the  multiplier  or  divisor  is  a  mixed  number,  reduce 
to  an  improper  fraction,  and  proceed  as  above. 

33.  Divide  8  lb.  11  oz.  15  pwt.  18  gr.  by  2|. 

34.  If  a  railroad  train  runs  60  mi.  35  rd.  16  ft.  in  one 
hour,  how  far  will  it  run  in  12|  hr.  at  the  same  rate  of  ' 
speed  ? 

35.  Divide  14  bu.  3  pk.  6  qt.  1  pt.  by  |. 

36.  Divide  5  yr.  1  mo.  1  wk.  1  da.  1  hr.  1  min.  1  sec.  by 
3|. 


MISCELLANEOUS   PROBLEMS.  161 

MISCELLANEOUS  PROBLEMS. 

220.  1«  Name  two  numbers  which  multiplied  together 
make  14. 

2.  Write  three  sets  of  factors  for  24. 

3.  Find  the  prime  factors  of  2205. 

14x6x3x2x8^^ 
•5x6x2x9x24"' 

5.  How  many  yards  of  silk,  24  inches  wide,  will  it  take 
to  line  a  skirt  containing  six  yards  of  cloth  28  inches  wide  ? 

6.  Find  the  least  common  multiple  of  2,  3,  4,  5,  6,  7, 

8,  9. 

7.  Find  the  greatest  common  divisor  of  285,  465. 

8.  Find  the  smallest  number  that  will  exactly  contain 

9,  15,  18,  20. 

9.  Find  the  length  of  the  longest  stick  that  will  exactly 
measure  the  sides  of  a  room  216  yd.  by  111  yd. 

10.  What  is  the  smallest  sum  of  money  with  which  you 
can  buy  pears  at  75^  a  basket,  peaches  at  90^,  and  grapes 
at  50^,  using  the  same  amount  of  money  for  each  kind  ? 

11.  How  many  times  is  1  contained  in  i  ? 

12.  How  many  times  is  \  contained  in  1  ? 

13.  A  man  bought  a  horse  for  $  240,  which  is  f  of  what 
he  sold  it  for.     What  did  it  sell  for  ? 

14.  A  man  bought  a  horse  for  f  240,  and  sold  it  for  4  of 
what  he  paid  for  it.     What  did  it  sell  for  ? 

15.  A  pole  stands  \  in  the  mud,  \  in  the  water,  and  the 
remaining  10  feet  are  above  the  water.  How  long  is  the 
pole? 

16.  A  man  owns  4  farms  containing  3651  375f,  284|, 
and  254-1  acres  respectively.     How  many  acres  in  all  ? 


162  MISCELLANEOUS   PROBLEMS. 

17.  The  man  owning  the  above  farms  sells  A  234/^  acres, 
and  B  366/^  acres.     How  many  acres  has  he  left  ? 

18.  What  is  the  value   of   3|  x  y^^  X  f  X  14  x  Sj^  x  ^ 

xAxfx^xi? 

19.  Find  the  least  common  multiple  of  273,  462,  1785, 
and  399. 

20.  A  man  owns  a  farm  containing  400  acres.  He  sells 
J  of  the  farm,  and  divides  the  remainder  among  his  six 
children.     How  many  acres  does  each  child  receive  ? 

21.  Find  the  sum  of  3  bu.  6  pk.  2  qt.  1  pt,  3  pk.  1  qt 
1  pt.,  7  bu.  3  pk.,  4  bu.  7  qt.  1  pt.,  and  19  bu.  2  pk.  2  qt. 
1  pt. 

22.  Find  the  sum  of  4  lb.  6  oz.  21  pwt.  9  gr.,  5  oz.  11  gr., 
3  lb.  9  oz.  18  pwt.,  11  oz.  17  pwt.  5  gr.,  16  lb.  4  oz.  11  pwt., 
18  lb.  17  gr.,  21  lb.  15  pwt.  11  gr.,  23  lb.  10  oz.  21  pwt. 
23  gr. 

23.  What  part  of  a  mile  is  214  rd.  2  yd.  2  ft.  3  in.  ? 

24.  Keduce  1  pk.  4  qt.  1|  pt.  to  the  fraction  of  a  bushel? 

25.  What  is  the  quotient  of  184  bales,  4  bundles,  1  ream, 
13  quires,  20  sheets,  divided  by  |  ? 

26.  A  farm  is  60  ch.  25  1.  long.  How  many  rods  long  is 
it? 

27.  A  surveyor  measured  my  farm,  and  found  that  it  is 
80  ch.  long  and  60  ch.  broad.  How  many  acres  does  it 
contain  ? 

28.  In  133128  in.  how  many  miles? 

29.  How  many  times  will  a  wheel  12  ft.  3  in,  in  circum- 
ference turn  round  in  going  15  mi.  20  rd.  12  ft.  2  in.  ? 

30.  How  many  steel  rails  30  ft.  long  are  needed  in  the 
construction  of  7  mi.  305  rd.  7  ft.  6  in.  of  double-track  rail- 
road? 


MrSCELLANEOUS   PROBLEMS.  163 

31.  What  fraction  of  the  year  is  contained  in  the  months 
of  July  and  August,  1896  ? 

32.  The  greatest  depth  of  the  Atlantic  telegraph  cable  is 
2  mi.  250  rd.  5  yd.  1  ft.     How  many  feet  deep  is  it  ? 

33.  How  many  statute  miles  in  45°  22'  30",  measured 
on  the  equator  ? 

34.  On  an  average,  when  walking,  Isaac  steps  24  inches 
twice  every  second.  How  many  minutes  will  it  take  him 
to  walk  1^  miles  ? 

35.  Reduce  -^-^  of  an  acre  to  square  rods  and  decimals  of 
a  square  rod. 

36.  A  silversmith  in  making  spoons  uses  2  lb.  3  oz.  19 
pwt.  of  silver  in  one  day,  3  lb.  18  pwt.  20  grs.  on  the  sec- 
ond day,  and  11  oz.  19  pwt.  23  gr.  on  the  third  day.  How 
much  silver  does  he  use  altogether  ? 

37.  What  decimal  of  a  pound  Troy  are  4  oz.  14  pwt.  ? 

38.  Reduce  18  pwt.  164  gr.  to  the  fraction  of  a  pound. 

39.  Find  the  difference  between  |  lb.  and  4  lb.  7.84  oz. 
Troy. 

40.  Find  the  cost  of  21  doz.  spoons,  each  weighing  9  oz. 
8.76  pwt.,  at  ^.045  a  pennyweight. 

41.  What  decimal  part  of  a  grain  is  j-^-^-^  of  a  pound  ? 

42.  AVhat  part  of  an  acre  is  f  sq.  rd.  ? 

43.  Multiply  16  bu.  9  pk.  8  qt.  by  |. 

44.  What  is  the  product  of  20  lb.  9  oz.  11  pwt.  15  gr. 
multiplied  by  |  ? 


164 


MEASUREMENTS. 


An  Angle. 


ME  A  S  U RE  ME  NTS, 

221.  An   Angle   is   the    difference    in    direction   of   two 
lines. 

222.  A  Right  Angle  is  the  angle  of 
a  square. 

223.  Anything  that  has  length  and 
breadth,  but  not  thickness,  is  a  Surface. 

224.  A  surface  that  does  not  change 
its  direction  is  a  Plane  Surface. 

225.  A  figure  having  four  straight 
sides  and  four  right  angles  is  a  Rect- 
angle. 

226.  A  Square  is  a  rectangle  having 

equal  sides. 


A  Right  Angle. 


A  Rectangle. 


227.  A  Square  Foot  is  a  square  1 
foot  long  and  1  foot  wide. 

228.  A  Square  Yard  is  a  square 
1  yd.  long  and  1  yd.  wide. 

229.  The  Area  of  a  surface  is  the 
number  of  square  units  that  it  con- 
tains. 

6  IN. 


z 


A  Square. 

Note,  — There  are  6  sq.  in. 
in  a  row,  and  in  4  rows  there 
are  4  times  6  sq.  in.  =24  sq.  in. 

The  multiplier  is  abstract, 
and  the  unit  of  the  product  must  be  the  same  as  the  unit  of  the 
multiplicand. 


MEASUREMENTS. 


165 


230.  The  lengtli  and  breadth  of  a  rectangle  are  called 
its  Dimensions. 

Length  x  Breadth  =  Area. 

Area  -r-  Length       =  Breadth. 

Area  -i-  Breadth      =  Length. 

231.  A  figure  having  three  straight  sides  and  three 
angles  is  a  Triangle. 

The  Base  of  a  triangle  is  the  line  upon  which  it  stands, 
and  the  Altitude  is  its  height  above  the  base,  or  the  base 
extended.  Thus,  AC  is  the  base,  and  BD  the  altitude,  of 
the  triangle  shown  below. 


(     It   is   evident    from  the   accompanying   figure    that    the 
Y  area  of  a  triangle  is  equal  to  one-half  the  area  of  a  rectangle 
\of  the  same  base  and  altitude. 

232.  Every  circle  may  be  regarded  as 
composed  of  many  equal  triangles,  the 
radius  of  the  circle  forming  the  alti- 
tudes, and  the  circumference  forming 
the  sum  of  the  bases.  Therefore,  the 
area  of  a  circle  is  equal  to  i  the  prod- 
uct of  the  circumference  and  radius. 

Principle.  —  The   circumference   of   a   circle   is   3.1416 
times  the  diameter,  or  about  3^. 


166  MEASUREMENTS. 

233.  Circumference  -7-  3.1416  =  Diameter. 

Diameter  x  3.1416  =  Circumference. 

Oral. 

Find  the  areas  of  rectangles  as  follows : 

1.  10  ft.  by  8  ft.  4.   50  ft.  by  20  ft. 

2.  16  ft.  by  4|-  ft.  5.    6  ch.  by  8  ch. 

3.  14  rd.  by  10  rd.  6.    9  yd.  by  6  yd. 

Find  the  other  dimension : 

7.  Area  24  sq.  ft.,  length  8  ft. 

8.  Area  72  sq.  yd.,  length  8  yd. 

9.  Area  100  sq.  in.,  breadth  5  in. 

10.  Length  16  yd.,  area  64  sq.  yd. 

11.  Breadth  4  ft.,  area  84  sq.  ft. 

12.  Area  56  sq.  ch.,  length  8  ch. 

Find  the  areas  of  the  following  triangles : 

13.  Base  10  ft.,  alt.  12  ft.        16.    Base  5  ft.,  alt.  10  ft. 

14.  Base  9  yd.,  alt.  6  yd.         17.    Base  12  in.,  alt.  8  in. 

15.  Base  15  in.,  alt.  6  in.         18.    Base  10  rd.,  alt.  5^  rd. 

Find  the  circumferences  of  circles  having  the  following 
diameters : 

Note.  —  Indicate  the  operation  only. 

19.  12  ft.  21.    16  in.  23.    16  rd.  25.    62  yd. 

20.  18  ft.  22.    14  yd.  24.    25  ch.  26.    84  ft. 

Find  the  diameters  having  the  following  circumferences : 
Note.  —  Indicate  only. 

27.    78  ft.      28.    19  ft.      29.    316.14  rd.      30.    189.68  ch. 


Fi^DiisG  arp:as.  167 

Written. 
Find  areas  : 

31.  Circumference  37.6992  ft.,  radius  6  ft. 

32.  Circumference  47.124  ft.,  diameter  15  ft. 

33.  Circumference  62.832  ft.,  diameter  20  ft. 

34.  Circumference  94.248  ft.,  diameter  30  ft. 

35.  Diameter  24  ft.  38.    Diameter  160  ft. 

36.  Radius  16  ft.  39.    Radius  62  ft. 

37.  Circumference  50  ft.       40.    Circumference  314.16  ft. 

41.  How  many  square  yards  are  there  in  a  floor  24  ft. 
long  and  15  ft.  wide  ? 

42.  The  base  of  a  triangle  is  20  ft.  and  the  altitude  18 
ft.     What  is  the  area  ? 

43.  The  circumference  of  a  circle  is  31.416  ft.  and  its 
radius  is  5  ft.     What  is  its  area? 

44.  When  the  diameter  of  a  circle  is  50  ft.,  what  is  the 
circumference  ? 

45.  When  the  radius  is  6  ft.,  what  is  the  circumference  ? 

46.  When  the  circumference  is  78.54  ft.,  what   is   the 
radius  ? 

47.  Mr.   Clark's  farm  is  35  ch.  long  and  25  ch.  wide. 
How  many  acres  does  it  contain  ? 

48.  A  certain  field  is  70  rd.  long  and  65  rd.  wide.     How 
many  acres  are  there  in  the  field  ? 

49.  A  piece  of  land  is  65  rd.  wide.     How  long  must  it 
be  to  contain  56^  acres  ? 

50.  How  many  sods  10  inches  square  will  be  required  to 
turf  a  lawn  100  ft.  long  and  50  ft.  6  in.  wide  ? 

51.  A  building  lot  measures  60  ft.  in  front.     What  must 
be  its  depth  to  contain  \  of  an  acre?        \^  \''i»^X'^  • 


168  MEASUREMENTS. 

^^^  52.    How  many  tiles,  each  8  in.  square,  will  be  required 
for  the  floor  of  a  room  24  ft.  by  30  ft.  ? 

53.    How  many  shingles  will  be  required  for  a  roof  45 
ft.  long,  and  each  of  its  two  sides  20  ft.  wide,  allowing  8 
shingles  to  the  square  foot  ? 
>      /    54.    How  many  square  yards  of  oil-cloth  are  needed  to 
L--^  cover  a  floor  18  ft.  by  24  ft.  6  in.  ? 

55.  A  owns  a  city  lot  168  ft.  long  and  42  ft.  wide.     He 
->/^         uses  f  of  it  for  a  lawn.     How  many  square  yards  does  the 

■  *     lawn  contain? 

56.  How  long  will  it  take  a  man  to  mow  the  above  lot,, 
if  it  takes  him  a  minute  to  run  a  2-foot  lawn-mower  length- 
wise of  the  lot  ? 

57.  A  pony  can  reach  40  feet  in  any  direction  from  the 
stake  to  which  he  is  picketed.  Over  how  many  square 
rods  of  surface  can  he  graze  ? 

58.  What  is  the  diameter  of  a  tree  that  is  10  ft.  in 
circumference  ? 

59.  A  basin  measures  9  in.  across  the  top  and  6  in. 
across  the  bottom.  How  much  farther  around  the  top 
than  around  the  bottom? 

60.  How  many  acres  of  land  are  enclosed  by  a  circular 
mile  track  ? 

61.  A  landscape  gardener  lays  off  a  circular  grass-plot 
whose  radius  is  one  rod,  and  near  it  a  semicircular  plot 
having  a  radius  two  rods  in  length.     Compare  their  areas. 

.         62.    If  the  area  of  a  triangle  is  9  acres  and  the  base  is 
\/  80  rods,  what  is  the  altitude  ? 

63.  Find  the  area  of  one  gable  end  of  a  building  40  ft. 
wide,  the  ridge  being  15  ft.  above  the  eaves. 

64.  Find  the  area  of  a  triangle  whose  altitude  is  14  ft. 
and  its  base  12  ft. 

65.  Find  the  area  of  a  triangle  whose  altitude  is  4  in.  and 
base  12  in. 


CABPETING   KOOMS*.  169 


CARPETING  ROOMS. 

234.  In  making  a  carpet,  the  carpeting  is  cut  from  a  roll 
into  strips  which  are  usually  laid  from  end  to  end  of  the 
floor,  or  lengthwise.  Sometimes  the  strips  are  laid  across 
the  room. 

1.  How  much  carpeting  must  I  purchase  to  cover  a  room 
6  yd.  long  and  4f  yd.  wide,  strips  running  lengthwise  ? 

Solution.  —  It  will  be  necessary  to  purchase  as  much  carpeting  as 
if  the  room  were  5  yd.  wide,  the  excess  of  \  yd.  being  turned  under  in 
the  last  strip. 

1  strip  contains  6  yd.     5  strips  =  5  times  6  yd.  =  30  yd.     Ans. 


2.  How  many  yards  must 
I  purchase,  if  the  strips  are 
laid  across  the  room  ? 

Solution.  —  1  strip  contains  4| 
yd.     6  strips  =  6  times  4f  yd.  = 

28-1-  yd,      ^j^s. 

Carpeting  is  commonly  1 
yd.  or  J  yd.  wide. 

Note.  —  It  is  often  necessary  to  6  yd. 

purchase  more  than  enough  carpet- 
ing to  cover  a  room,  on  account  of  the  waste  in  matching  patterns. 

This  diagram  represents  the  &oor  in  Example  1,  in  which 
the  strips  are  laid  lengthwise.  Pupils  should  draw  a  similar 
diagram  for  each  floor. 

Note.  —  Carpeting  is  sold  by  lineal  yards  or  metres,  not  by  square 
measure. 

3.  A  merchant  bought  a  roll  of  carpet  containing  74  yd. 
at  85^  a  yard,  and  sold  it  at  $  1.15  a  yard.  What  was  his 
profit  ? 

4.  A  roll  of  carpet  f  yd.  wide  contains  60  yards.  How 
many  square  yards  of  surface  will  it  cover  ? 


170  MEASUREMENTS. 

5.  How  many  strips  of  carpet  3  ft.  wide  will  cover  a 
floor  15  ft.  wide  ?     17  ft.  wide  ?     18  ft.  wide  ? 

6.  How  many  strips  27  in.  wide  are  required  for  a  floor 
12  ft.  wide  ?     14  ft.  wide  ?     16  ft.  wide  ? 

7.  If,  in  Example  5,  the  room  is  16  ft.  long,  how  many 
linear  yards  of  carpet  will  be  needed  to  cover  the  floor  ? 

8.  If,  in  Example  6,  the  room  is  19  ft.  long,  how  many 
yards  will  be  required  to  cover  the  floor  ? 

9.  How  many  yards  of  ingrain  carpet  J  yd.  wide  will  be 
required  for  a  floor  17  ft.  wide  and  20  ft.  long,  strips  run- 
ning across  the  room  ? 

10.  How  much  will  a  carpet  cost  at  $  .90  a  yard  to  cover 
a  floor  22  ft.  long  and  15  ft.  wide,  if  the  strips  run  crosswise, 
and  no  allowance  is  made  for  matching  ? 

11.  How  many  yards  of  carpet  f  yd.  wide  will  be  re- 
quired for  a  floor  20  ft.  long  and  15  ft.  wide,  if  the  strips 
run  across  the  room  ? 

12.  How  many  yards  of  carpet  1  yd.  wide  will  be  re- 
quired for  a  floor  18  ft.  long  and  14  ft.  wide,  strips  running 
across  the  room  ? 

13.  How  many  yards  of  carpeting  are  needed  to  cover 
a  floor  24  ft.  long  and  17  ft.  wide,  strips  running  lengthwise 
and  I  yd.  wide  ? 

14.  If  my  room  is  16^  ft.  long  and  12  ft.  wide,  how  many 
yards  of  carpeting  24  inches  wide  must  I  buy,  if  in  cutting 
6  inches  is  allowed  on  each  strip  for  matching  ? 

15.  I  wish  to  have  a  carpet  woven.  My  room  is  21  ft. 
long  and  17  ft.  wide.  How  much  carpeting,  34  inches  wide, 
must  I  order  to  exactly  cover  the  room,  no  allowance  being 
made  for  matching  ? 


PLASTEKINCi    AND    PAINTING.  171 

16.  How  many  yards  of  carpet  2^  ft.  wide  will  cover  a 
floor  7^  yd.  long  and  14  ft.  wide,  if  strips  run  length- 
wise, and  it  requires  ^  yd.  for  matching  ? 

17.  What  will  it  cost  to  carpet  a  room  15  ft.  by  17-^  ft., 
with  carpet  30  in.  wide,  at  $1.20  per  lineal  yard,  if  the 
strips  run  lengthwise,  and  an  allowance  of  9  in.  to  each 
strip  be  made  for  matching? 

18.  How  much  less  would  be  the  cost  with  no  loss  for 
matching?     (Ex.  17.) 

19.  A  room  31  ft.  by  17  ft.  is  to  be  covered  with  carpet- 
ing 30  in.  wide.  How  many  yards  must  be  purchased,  and 
how  wide  a  strip  must  be  turned  under  ? 

20.  At  $  2.50  a  yard,  what  will  be  the  cost  of  a  carpet 
to  cover  a  parlor  floor  6  yd.  long  and  5^  yd.  wide,  if  |  yd. 
is  wasted  in  matching  ? 

21.  How  many  yards  of  matting  li  yd.  wide  will  be 
required  for  an  assembly  room  85  ft.  8  in.  long  and  64  ft. 
6  in.  wide,  strips  running  across  the  room  ? 

22.  A  room  17  ft.  6  in.  long,  14  ft.  wide,  is  to  be  car- 
peted with  carpet  J  yd.  wide.  A  border  -|  yd.  wide  goes 
around  the  outside.  How  many  yards  of  border,  and  how 
many  yards  of  carpet,  if  strips  run  lengthwise,  and  there 
is  a  waste  of  one  foot  on  each  strip  for  matching? 

PLASTERING  AND  PAINTING,  ETC. 

235.  Plastering  and  painting  are  usually  done  by  the 
square  yard.  Allowance  is  sometimes  made  for  doors  and 
windows,  which  are  called  openings.  Allowance  is  also 
sometimes  made  for  base-boards  and  wainscoting? 

1.  A  room  is  18  ft.  long,  12  ft.  wide,  and  10  ft.  high. 
How  many  square  yards  in  the  walls  and  ceiling,  making 
no  allowance  for  openings  ? 


172  MEASUREMENTS. 

Note.  —  Let  the  pupils  draw  a  diagram  for  each  room,  represent- 
ing the   four  walls  in   a  line.      The   entire 

length  of  the  walls  will  be  2    x  (18  ft.    +  ^o  ft 

12  ft.)  =  60  ft.     The  area  of  the  four  walls,  ; """j 

60  ft.  by  10  ft.  =  600  sq.  ft.  I                   | 

Area  of  ceiling  18  ft.  by  12  ft.  =216  sq.  ft.  -'1                  • 

600  sq.  ft.  +216  sq.  ft.  =  816  sq.  ft.  =  90|  ^1                   \ 


,  area  of  ^ 

svalls  and  ceilmg.                             ;                  i 

j                  1 

. 

:              ;                    1 

««-i 

•    1             '                   I 

o 

'             1                    1 

18  a.  .12.11.  .18  ft.  12  ft. 

2.  Find  the  cost  of  plastering  the  walls  and  ceiling  of 
a  room  35  ft.  long,  26  ft.  6  in.  wide,  and  15  ft.  high,  at  $  .45 
a  square  yard,  allowing  1024  sq.  ft.  for  doors,  windows,  and 
base-board  ? 

3.  How  many  square  yards  of  plaster  in  the  sides  and 
ceiling  of  a  room  30  ft.  long,  24  ft.  wide,  and  10  ft.  high, 
allowing  for  a  base-board  1  ft.  high,  2  doors  3  ft.  by  8  ft., 
and  4  windows  3  ft.  by  6  ft.  ? 

4.  Find  the  cost  of  plastering  the  ceiling  of  a  room  18 
ft.  by  20  ft.,  at  10  cents  a  square  yard. 

5.  A  room  15  ft.  by  18  ft.,  and  10  ft.  high,  has  4  doors 
each  3  ft.  by  7  ft.,  and  3  windows  each  3  ft.  by  6  ft. 
Find  the  cost  of  plastering  the  walls  and  ceiling  of  the 
room  at  30  cents  a  square  yard. 

6.  How  many  square  yards  of  plastering  in  the  ceiling  of 
a  room  20  ft.  long,  9  ft.  high,  and  15  ft.  wide,  no  allowance 
for  openings  ? 

7.  At  $  .30  a  square  yard,  how  much  will  it  cost  to  plaster 
a  room  21  ft.  6  in.  long,  16  ft.  wide,  and  9  ft.  high,  the 
base-board  being  8  in.  wide,  and  allowing  for  3  windows 


PAPERING   WALLS.  173 

7  ft.  by  2^  ft.,  and  6  doors  of  the  same  dimensions  as 
the  windows  ? 

8.  Find  the  cost  of  plastering  the  walls  and  ceiling  of 
a  room  which  is  36  ft.  long,  27  ft.  wide,  and  9  ft.  high,  at 
25i^  per  square  yard. 

9.  My  study  is  18  ft.  long,  16  ft.  wide,  8i  ft.  high,  and 
contains  1  door  3  ft.  by  7  ft.,  and  2  windows,  each  3  ft.  by 
6  ft.  The  base-board  is  9  in.  high.  What  will  it  cost, 
at  36^  per  square  yard,  to  plaster  it,  making  full  deduction 
for  openings  ? 

10.  At  35  cents  a  square  yard,  what  will  be  the  cost  of 
plastering  the  walls  and  ceiling  of  a  room  6  yd.  long, 
5  yd.  wide,  and  3  yd.  high,  an  allowance  of  20  sq.  yd. 
being  made  for  openings,  etc.  ? 

11.  Find  the  cost  of  plastering  a  room  18  ft.  square  and 
10  ft.  high,  at  25  cents  a  square  yard,  ^  being  deducted  for 
openings  ? 

12.  A  close  fence  6i  ft.  high  surrounds  a  vacant  lot  450 
ft.  by  380  ft  At  7  cents  a  square  yard,  what  will  be  the 
cost  of  painting  both  sides  of  the  fence  ? 

13.  Find  the  cost  at  18j^  per  square  yard  to  plaster  the 
sides  and  bottom  of  a  cistern  8  ft.  6  in.  square,  and  9  ft, 
deep. 

14.  Find  the  square  yards  of  plastering  on  a  room  20  ft, 
long,  17  ft.  6  in.  wide,  9  ft.  high.  Allow  for  6  windows, 
each  7  ft.  6  in.  high,  3  ft.  wide,  and  4  doors,  each  7  ft, 
high  and  3  ft.  9  in.  wide. 

PAPERING    WALLS. 

236.   Wall-paper  is  sold  by  the  roll.     A  Single  Roll  is 

8  yd.  long,  a  Double  Roll,  16  yd.  long.     Borders  are  sold 
by  the  lineal  yard. 


174  MEASUREMENTS. 

The  number  of  rolls  needed  for  a  room  is  found  by 
dividing  the  area  of  the  space  to  be  papered  by  the  area  of 
one  roll.     The  width  of  wall-paper' is  commonly  18  in. 

Notes. — Unless  otherwise  stated,  a  roll  is  considered  as  8  yd. 
long  and  18  in.  wide. 

Dealers  in  wall-paper  do  not  sell  a  part  of  a  roll.  If  a  part  of 
a  roll  is  needed,  a  whole  roll  must  be  purchased. 

1.  What  would  it  cost  to  paper  the  walls  of  a  room  18 
ft.  long,  12  ft.  wide,  and  9  ft.  high,  with  paper  8  yd.  to 
the  roll,  and  i  yd.  wide,  at  45^  a  roll  ? 

2.  How  many  strips  of  paper,  and  how  many  double 
rolls,  will  paper  the  sides  of  a  room  15  ft.  long,  12  ft.  wide, 
and  8  ft.  high,  each  roll  being  1^  ft.  wide  and  16  yd.  long, 
no  allowance  being  made  for  matching  ? 

3.  How  many  double  rolls  of  paper  16  yd.  to  the  roll, 
■1-  yd.  wide,  will  be  required  to  paper  the  walls  and  ceiling 
of  a  room  25  ft.  long,  10  ft.  wide,  and  10  ft.  high,  110 
sq.  feet  being  deducted  for  doors,  windows,  etc.  ? 

4.  How  many  rolls  of  paper  will  be  required  for  the 
walls  of  a  room  16  ft.  by  20  ft.,  and  9  ft.  high  above  the 
base-board,  allowing  for  3  doors,  each  3  ft.  by  7  ft.,  and  3 
windows,  each  3  ft.  by  6  ft.  ? 

5.  What  will  be  the  cost  of  the  paper  and  border  for 
the  above  room  at  30  cents  a  roll  for  the  paper,  and  15 
cents  a  yard  for  the  border  ? 

6.  How  many  rolls  of  paper  must  be  purchased  to 
paper  the  walls  and  ceiling  of  a  library,  12  ft.  long,  10  ft. 
6  in.  wide,  and  8  ft.  high,  the  base-board  being  6  in.  wide  and 
the  border  1^  ft.  wide,  with  paper  ^  yd.  wide  and  8  yd.  long, 
the  paper  extending  from  the  border  to  the  base-board '/ 


BOARD   MEASURE.  175 

BOARD  MEASURE. 

237.  A  Board  Foot  is  a  square  foot  of  the  surface  of  a 
board,  1  inch  thick,  or  less. 

To  find  the  number  of  board  feet  in  lumber  that  is  more 
than  one  inch  thick,  we  must  multiply  the  number  of  board 
feet  in  the  surface  by  the  number  of  inches  in  the  thickness. 

A  board  10  ft.  long,  1  ft.  wide,  and  1  in.  thick  or  less  con- 
tains 10  board  feet;  but  a  beam  10  ft.  long,  1  ft.  wide,  and 
8  in.  thick  contains  8  times  10  board  feet  =  80  board  feet. 

To  find  the  number  of  board  feet  in  a  tapering  board, 
the  average  width  must  be  found  by  taking  i-  the  sum  of 
the  widths  of  the  two  ends.  Thus  a  board  10  ft.  long,  and 
12  in.  wide  at  one  end,  and  6  in.  wide  at  the  other,  contains 
as  many  board  feet  as  if  it  had  a  uniform  width  of  9  in. 
(12  4-  6)  -f-  2  =  9. 

238.  The  number  of  board  feet  =  Length  (in  feet)  x 
Width  (in  feet)  x  Thickness  (in  inches). 

Note.  —  When  the  thickness  is  one  inch  or  less,  the  number  of 
board  feet  is  the  product  of  the  length  and  width  in  feet. 

1.  How  many  board  feet  in  a  board  15  ft.  long,  15  in. 
wide,  and  1  in.  thick  ? 

2.  How  many  board  feet  would  there  be  in  the  board 
(Ex.  1)  if  it  were  |  in.  thick  ?     2  in.  thick  ?     1 1-  in.  thick  ? 

3.  How  many  feet  of  lumber  one  inch  thick  will  be 
required  for  a  tight  board  fence  6  ft.  high  around  a  yard 
4  rods  square  ? 

4.  How  much  lumber  (Ex.  3)  will  be  required  for  an 
open  board  fence,  4  boards  high,  boards  8  in.  wide,  and  5 
in.  apart  ? 

5.  I  need  213  planks  4  ft.  8  in.  long,  1  ft.  wide,  and  2 
in.  thick,  to  build  a  sidewalk.  How  much  will  they  cost  at 
f  13  a  thousand  ? 


176  MEASUREMENTS. 

6.  How  many  feet  of  lumber  will  it  take  to  build  a  line 
fence  168  ft.  long,  the  fence  being  5  boards  high,  and  the 
boards  6  in.  wide  ? 

7.  What  will  be  the  cost  of  10  planks,  each  12  ft.  long, 
10  in.  wide,  and  3  in.  thick,  at  $  16  per  M.  ? 

8.  Find  the  cost  of  a  stick  of  timber  8  in.  square,  and 
40  ft.  long,  at  $  18  per  M. 

9.  What  is  the  cost  of  8  sticks  of  timber  each  36  ft. 
long,  10  in.  wide,  8  in.  thick,  at  $  12  per  M.  ? 

10.  How  many  board  feet  of  2-inch  planking  will  it  take 
to  make  a  walk  4  feet  long  and  3  feet  wide  ? 

11.  rind  the  cost  of  7  planks  12  ft.  long,  16  in.  wide  at 
one  end,  and  12  in.  at  the  other,  at  $  .08  a  board  foot. 

12.  At  f  18  per  M.,  find  the  cost  of  flooring  a  room  21 
ft.  by  16  ft.,  allowing  i  of  the  lumber  for  matching.' 

Note.  — Find  area  of  floor  and  add  |. 

13.  Find  the  cost  of  a  board  20  ft.  long,  22  in.  wide  at 
one  end,  and  tapering  to  16  in.  at  the  other,  and  1^  in. 
thick,  at  $30  per  M.? 

14.  At  $12  per  M.,  what  will  be  the  cost  of  2-inch  plank 
for  a  3  ft.  6  in.  sidewalk  on  the  street  side  of  a  rectangular 
corner  lot  56  ft.  by  106  ft.  6  in.  ? 

MISCELLANEO  US. 

239.  1.  My  dining-room  is  15  ft.  long  and  12  ft.  wide; 
the  walls  are  10  ft.  high.  What  will  it  cost  to  paper  the 
walls  and  ceiling  with  paper  1\  ft.  wide,  if  there  are  8  yd. 
in  a  roll,  and  each  roll  costs  $.37|.  (J^  allowed  for 
openings.) 

2.  What  will  a  carpet  for  the  dining-room  (Ex.  1)  cost 
me  at  $  .75  a  yd.,  carpet  j  yd.  wide  ? 


MISCELLANEOUS.  177 

3.  There  are  three  windows  in  the  dining-room.  What 
will  it  cost  to  furnish  them  with  shades  at  $  1.10  each  and 
sash-curtains  at  $  1.37^  each  ? 

4.  I  bought  a  table  at  $  14.50,  six  chairs  at  $  2.75  each, 
a  sideboard  for  f  30,  and  other  furniture  for  $28.97.  I 
also  spent  $  20  for  new  table  linen.     What  did  it  all  cost  ? 

5.  What  was  the  entire  cost  of  refurnishing  my  dining- 
room  ?     (Ex.  1,  2,  3,  4.) 

6.  What  will  it  cost  to  carpet  a  room  which  is  24  ft. 
long,  and  18  ft.  wide,  with  Brussels  carpet  1  yd.  wide,  no 
waste  in  matching,  at  $  1  per  yard  ? 

7.  Find  the  cost  of  plastering  sides  and  ceiling  of  a  room 
26  ft.  long,  13|-  ft.  wide,  13  ft.  high,  at  9  cents  a  square 
yard,  allowing  25  sq.  yd.  for  openings. 

8.  What  will  it  cost  to  build  a  cement  walk  40  ft.  long 
and  6  ft.  wide,  at  $  1.25  per  square  yard  ? 

9.  A  field,  containing  8  acres,  is  60  rd.  long.  How  wide 
is  it  ? 

10.  How  many  yards  of  carpet  will  cover  a  floor  18  ft. 
long,  16  ft.  wide?  Carpet  one  yard  wide,  strips  to  run 
lengthwise  of  room. 

How  many  yards  if  the  strips  run  crosswise  of  the  room  ? 

11.  Find  the  cost  of  fencing  a  rectangular  corner  lot  68 
ft.  by  130  ft.,  the  street  fence  costing  54  cts.  a  yard,  and 
the  line  fences  25  cents  a  yard,  but  only  half  of  the  cost  of 
the  latter  to  be  charged  to  the  lot. 

12.  Find  the  cost  of  a  carpet  f  of  a  yd.  wide,  at  $  1.50 
per  lineal  yard,  for  a  room  20  ft.  long  and  18  ft.  wide, 
strips  running  lengthwise,  and  allowing  a  waste  of  ^  of  a 
yard  on  each  strip  for  matching. 

13.  What  is  the  breadth  of  a  rectangular  lot  whose  area 
is  75  sq.  ch.  and  the  length  9  ch.  ? 


178  MEASUREMENTS. 

14.  What  is  the  circumference  of  a  circle  whose  radius 
is  9  ft.  ? 

15.  How  many  square  yards  in  the  above  circle  ?  (Ex.  15.) 

16.  How  many  revolutions  does  the  5-foot  driving-wheel 
of  a  locomotive  make  in  going  30  miles  ? 

17.  Find  the  area  of  a  triangle  whose  base  is  5  ft.  and 
altitude  3  ft. 

18.  If  the  circumference  of  the  earth  is  25000  miles, 
what  is  the  diameter  ? 

19.  The  circumference  of  a  circle  is  18  ft.  What  is  its 
radius  ? 

20.  How  many  square  yards  in  a  triangle  whose  base  is 
48  ft.  and  whose  altitude  is  24  ft.  ? 

21.  Find  the  area  of  the  gable-end  of  a  house  whose 
width  is  25  feet  and  whose  ridge  is  10  feet  6  inches  higher 
than  the  base  of  the  gable. 

22.  If  the  diameter  of  the  earth  is  8000  miles,  what  is 
the  circumference  ? 

23.  How  many  board  feet  in  10  planks  18  ft.  long,  15 
in.  wide,  and  2  in.  thick,  and  what  will  they  cost  at  $40 
per  M.  ? 

24.  Find  the  cost  of  10  joists,  3  in.  by  12  in.,  16  ft.  long, 
at  $  25  per  M.  ? 

VOLUMES. 

240.  Anything  that  has  length,  breadth,  and  thickness  is 
called  a  Solid  or  Volume. 

241.  A  Rectangular  Volume  is  a  solid  having  six  rectan- 
gular faces. 

242.  A  Cube  is  a  solid  having  six  equal  square  faces. 

243.  A  Cubic  Inch  is  a  cube  1  inch  long,  1  inch  wide,  and 
1  inch  thick. 


VOLUMES. 


179 


m. 


top 


3  in.  wide 


244.   The  Volume  or  Solidity  of  a  body  is  the  number  of 
cubic  units  that  it  contains. 

1.   How  many  cubic  inches  in  a  block  4  in.  long,  3 
wide,  and  2  in.  thick  ? 

The  block  is  made  up  of  two  layers,  each  1  inch  thick.     In  the 
layer  there  are  4   times  3  cu.   in.     In 
the    two    layers,    therefore,    there    are 
2  X  (4  X  3  cu.  in.)  =  24  cu.  inches. 

The  multiplier  must  be 
considered  as  abstract. 

The  three  dimensions 
must  have  the  same  unit. 
The  length,  breadth, 
and  thickness  of  a  rec- 
tangular solid  are  its  di- 
mensions. 


3  cu.  in. 
12  cu.  in. 
24  cu.  in. 


K.  — v-^^y^ 

^ 

^^ 

[k  x.A-^§< 

milir-~^"~   ^ 

lilt  N 

-  V \ 

N~^ 

l|r 

'i! 

l^J 

™.:iiiii 

245.    Length  x  breadth  x  thickness  =  Solidity. 
Solidity  -j-  either   dimension  =  the  product  of   the  other 
two. 

Solidity  -i-  the  product  of  two  dimensions  =  the  other. 

2.  Find  the  number  of  cubic  feet  of  air  in  a  schoolroom 
32  ft.  square  and  12  ft.  high. 

3.  How  high  is  a  room  that  is  24i  ft.  long,  20  ft.  wide, 
and  contains  4410  cu.  ft.  ? 

4.  A  cubic  foot  of  ice  weighs  56\  pounds.  How  much 
will  a  load  of  22  cakes  weigh,  each  cake  measuring  2  ft. 
square  and  1  ft.  thick  ? 

5.  The  capacity  of  a  rectangular  box  is  480  cu.  in.  The 
box  is  8  in.  wide  and  5  in.  deep.     How  long  is  it  ? 

6.  A  schoolroom  is  25  ft.  long,  18  ft.  wide,  and  12  ft. 
high.  If  60  pupils  are  seated  in  it,  how  many  cubic  feet 
of  air  are  allowed  for  each  child  ? 

7.  A  man  sold  3  blocks  of  Vermont  marble,  each  8  ft. 
long,  and  6  in.  x  6  in.  at  the  ends.  How  much  did  he 
receive  for  the  marble  at  $  3.50  per  cubic  foot  ? 


180  MEASUREMENTS. 

y  8.  A  hot-house  bed  is  3  ft.  9  in.  long,  and  3  ft.  4  in. 
wide,  inside  measure.  How  deep  must  it  be  to  contain  25 
cu.  ft.  of  earth,  and  allow  6  in.  for  the  growth  of  the  plants  ? 

9.  How  many  bricks  8  in.  by  4  in.  and  2  in.  thick  will  be 
needed  for  a  wall  60  ft.  long,  20  ft.  high,  and  2  ft.  thick, 
making  no  allowance  for  mortar  ? 

10.  How  many  rectangular  blocks  12  in.  by  8  in.  by  3  in. 
can  be  packed  into  a  wagon-box  10  ft.  long,  4  ft.  wide,  and  2 
ft.  6  in.  deep  ? 

11.  How  many  cubic  yards  of  earth  must  be  excavated 
from  a  cellar  30  ft.  long,  21  ft.  wide,  and  5  ft.  8  in.  deep? 

12.  How  many  square  feet  in  the  surface  of  a  rectangular 
box  3  ft.  4  in.  long,  2  ft.  2  in.  wide,  and  li  ft.  high  ? 

13.  How  many  cubes  2  inches  on  each  edge  can  be 
sawed  from  a  block  of  marble  10  ft.  2  in.  long,  6  ft.  5  in. 
wide,  and  3  ft.  4  in.  thick  ? 

14.  A  box  is  1.5  in.  long,  .85  in.  wide,  and  .58  in.  deep. 
What  is  its  capacity  in  cubic  inches  ? 

15.  The  altitude  of  a  cylinder  is  8  ft.  and  the  circum- 
ference of  the  base  is  3  ft.  What  are  the  cubic  contents 
of  the  cylinder  ? 

Notes.  —  Contents  of  a  cylinder  =  Area  of  Base  x  Altitude. 

Area  of  curved  surface  of  a  cylinder  =  Circumference  of 
Base  X  Altitude. 
This  may  be  seen  by  cutting  a  piece  of  paper  so  that  it  will  exactly 
cover  the  curved  surface  of  a  small  cylinder. 

16.  What  is  the  area  of  the  curved  surface  in  the  cylinder 
mentioned  in  Example  15  ? 

17.  How  much  tin  will  be  required  to  make  2  doz.  cylin- 
drical shaped  cans,  with  a  diameter  of  4  in.  and  altitude 
of  7  in.,  allowing  tin  for  the  curved  surface  and  the  two 
circular  ends  ? 


WOOD  MEASURE.  181 

18.  How  many  cubic  inches  of  water  will  the  2  doz.  cans 
(Ex.  17)  contain  ? 

19.  What  will  it  cost  to  dig  a  cellar  36  ft.  long,  24  ft. 
wide,  and  6  ft.  deep,  at  20  cts.  a  cubic  yard  ? 

WOOD  MEASURE. 

246.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet 
high  makes  a  Cord. 

One  of  the  8  feet  in  length  of  a  cord  of  wood  is  a  Cord 
Foot. 

Note.  —  This  may  be  illustrated  by  placing  side  by  side  8  books  of 
equal  size.     One  of  the  books  represents  a  cord  foot. 

How  many  cords  of  wood  in  the  following : 

1.  A  pile  18  ft.  long,  4  ft.  wide,  8  ft.  high? 

2.  A  pile  50  ft.  long,  8  ft.  wide,  6  ft.  high? 

3.  A  pile  19  ft.  long,  2  ft.  wide,  5^  ft.  high  ? 

4.  A  pile  16  ft.  long,  4i  ft.  wide,  7  ft.  high? 

5.  What  is  the  cost  of  a  pile  of  wood  10  ft.  long,  4  ft. 
wide,  and  8  ft.  high,  at  $  4-J-  a  cord  ? 

6.  How  high   must   a   pile  of  4-foot  wood  be  piled  to 
contain  10  cords,  if  the  pile  is  50  ft.  long  ? 

7.  How  many  cords  of  wood  can  be  piled  in  a  shed  24  ft. 
long,  18  ft.  wide,  and  12  ft.  high  ? 

8.  How  many  cords  of  building-stone  in  a  pile  18  ft.  long, 
6^  ft.  wide,  and  3  ft.  high  ? 

9.  At  $  3.50  a  cord,  what  will  be  the  cost  of  a  pile  of 
stone  15  ft.  long,  4i  ft.  wide,  and  5  ft.  high  ? 

10.  How  many  cubic  feet  in  a  cord  of  2-foot  wood? 
3-foot  wood  ?     18-inch  wood  ? 


182  MEAStTREMBNTS. 


CAPACITY  OF  BINS. 


247.  1.  A  bushel  fills  2150.42  cubic  inches  of  space. 
How  many  bushels  of  wheat  can  be  contained  in  a  bhi  5 
ft.  X  5  ft.  X  4  ft.  ? 

5  X  5  X  4  X  1728  -  2150.42. 

Note.  —  A  bushel  fills  1^  cu.  ft.  of  space  nearly. 

2.  A  wine  gallon  fills  231  cubic  inches  of  space.  How 
many  gallons  of  water  can  be  contained  in  a  rectangular 
tank  10  ft.  by  8  ft.  by  4  ft.  ? 

Note.  —  A  cubic  foot  of  space  contains  y/jS  gal.  =  7 1  gal.  nearly. 

Find  the  contents  in  bushels : 

3.  Of  a  bin  6  ft.  long,  5  ft.  wide,  and  4  ft.  higl>. 

4.  Of  a  wagon-box  10  ft.  long,  42  in.  wide,  and  22  in.  high. 

5.  Of  a  box  3  ft.  by  21  ft.  by  2i  ft. 

6.  How  high  must  a  bin  8  ft.  long  and  5  ft  wide  be  built 
to  contain  120  bushels  ? 

Find  the  contents  in  gallons  : 

7.  Of  a  tank  8  ft.  by  6  ft.  by  2^  ft. 

8.  Of  a  cistern  6  ft.  by  5  ft.  by  41  ft. 

9.  Of  a  tank  5 J  ft.  square  and  6  ft  deep. 

10.  How  many  barrels  of  water  will  a  cistern  contain  that 
is  6  ft.  by  6  ft.  by  7  ft.  ? 

11.  A  circular  cistern  is  5  ft.  in  diameter  and  6  ft.  deep. 
How  many  barrels  of  water  will  it  hold  ? 

Note.  —  Area  of  base  x  altitude. 

12.  How  deep  must  I  build  a  bin  that  is  6  ft.  square,  to 
hold  90  bushels  of  wheat  ? 

13.  How  deep  must  I  build  a  tank  that  is  5  ft.  square  to 
hold  40  barrels  ? 


LONGITUDE  AND   TIME. 


248.  A  Meridian  is  an  imaginary  line  running  from  the 
north  pole  to  the  south  pole. 

All  places  on  a  meridian  have  the  same  time. 

Note.  —  The  meridians  of  Greenwich  and  Washington  are  the 
meridians  that  run  through  Greenwich  and  Washington. 

249.  Longitude  is  distance  east  or  west  from  some  stand- 
ard meridian,  as  Greenwich  or  Washington.  When  two 
places  are  on  the  same  side  of  the  standard  meridian,  their 
difference  in  longitude  is  found  by  subtraction.  When  on 
opposite  sides,  their  difference  in  longitude  is  found  by 
addition. 

1.  What  is  the  difference  in  longitude  between  two  cities, 

one  of  which   is  20°  west   longitude,  the  other  30°   east 

longitude  ? 

20°  +  30°  =  50°.     Ans. 

2.  What  is  the  dijfference  in  longitude  between  two  places, 
one  of  which  is  40°  E.,  the  other  70°  E.  ? 

70°  _  40°  =  30°.     A71S. 

Note.  —  No  two  places  can  have  a  difference  in  longitude  exceed- 
ing 180°.  If,  in  finding  difference  in  longitude  by  addition,  the  sum 
exceeds  180°,  subtract  the  sum  from  360°  to  lind  the  true  difference. 

The  earth  turns  upon  its  axis  from  west  to  east  once  in 
24  hours,  thus  -^^  of  its  entire  circumference,  or  15°  of  lon- 
gitude, passes  under  the  sun  in  1  hour. 

Since  the  earth  turns  at  the  rate  of  15°  every  hour,  in 

183 


184 


LONGITUDE   AND   TIME. 


1  minute  it  turns  ^^^  of  15°,  or  15',  and  in  1  second  ^  of  15', 
or  15".     Hence, 

The  earth  rotates  15°  in  i  hour,  15'  in  i  minute,  and  15" 
in  I  second. 

3.    The  difference  in  longitude  between  two  cities  is  18° 
30'.     What  is  the  difference  in  time  ? 


15  I  18^ 


30' 


Solution.  —  Since  the  earth  turns  15°  in 
1  hr.,  15'  in  1  min,,  15"  in  1  sec,  the  time 
1  hr.  14  min.     can   be   found  by  dividing  the  number  of 
degrees,  minutes,  and  seconds  by  15. 

4.    The  difference  in  time  between  two  cities  is  54  min. 
19  sec.     What  is  their  difference  in  longitude  ? 

Since  the  earth  turns  15°  in  1  hr,,  15'  in 

54  mm.  ly  sec.      i  min.,  and  15"  in  1  sec,  the  distance  in 

15  degrees,  minutes,  and  seconds  may  be  found 


13°  34'  45"             ^y  multiplying  the  number  of  hours,  mi 
and  seconds  by  15. 

DateUneNoT^S 

6  AJM.                                                                ^ 

507.32  7  A.M.  166°         ^^ 
8AJI.   150°,---'r' 

°              165°    6  A.M. 

T'^-.v^^^^  150°  4  A.M. 

BA.M.  ISBV/"^    \           \ 

/           /         N.     135°  3A.M. 

10A,M.  120=/          \^      \        \ 

/       /         /            \    120°  2  A.M. 

llAJI    106°/^       ^^\?\\\ 

\  / /    ^^^                J\  106°     1A.M. 

'^i~- 

'          ^^^ 

Nov.23 
90"  12  Midnight 
Nov.2a 

1         ^-^ 

3  PJM.  75  "Y"-              ^.■'^^/W^^/ 

\V\^\^^     ^"""~~-y  76°nPJtt. 

2  P.M.  eo°V^        y^     /    / 

\      \          ^v             y    80°  10  P.M. 

8  P.M.  45X.            /          / 

\          \       ^/45°9PJI., 

*PJL  30^^-->^,/____^^ 

\,.,^^^*ii3S^ 

6PJL   15°             g^              16'7PJt. 

6P 

Eiimel 

Nc 

.M. 

kleridlan 

T.22 

Diagram  showing  the  Difference  in  Time  at  Different  Meridians. 


STANDARD  OR  RAILROAD  TIME.  185 

Let  the  figure  on  page  184  represent  the  earth  rotating  on 
its  axis.  When  it  is  6  p.m.,  November  22,  on  the  prime 
meridian,  it  is  noon  of  the  same  day  at  90°  west  longitude, 
and  midnight  at  90°  east  longitude.  It  is  therefore  a.m. 
above  90°  west  longitude  and  p.m.  below. 

How  do  we  know  where  it  is  p.m.  and  where  a.m.  at  a 
given  place  on  the  earth's  surface  ? 

How  long  does  it  take  a  given  point  on  the  earth's  surfa\}e 
to  move  once  around  its  circle  ?  | 

Through  how  many  degrees  does  the  earth  rotate  in  one 
hour  ?     In  one  minute  ?     In  one  second  ? 

The  student  will  notice  that  in  travelling  around  the  earth 
from  west  to  east  there  is  an  apparent  gain  of  one  day,  since 
in  this  case  the  traveller  is  going  with  the  sun ;  in  going 
around  from  east  to  west  there  is  an  apparent  loss  of  one 
day. 

From  the  above  considerations  we  see  that  there  must  be 
somewhere  a  line  at  which  November  23,  and  in  general  each 
new  day,  must  begin  on  the  earth.  This  line  is  taken  180° 
from  the  prime  meridian,  and  is  called  the  date  line.  On 
crossing  this  line,  ships  either  gain  or  lose  a  day,  and  must 
correct  their  calendar  accordingly. 

Is  it  possible  to  have  a  day  last  48  hours  ?  Which  way  is 
the  ship  sailing  if  such  is  the  case  ? 

If  a  person  were  to  travel  around  the  earth  from  east  to 
west  in  120  days,  reckoning  by  local  time  in  various  places, 
in  how  many  days  would  he  actually  make  the  trip  ? 

STANDARD   OR  RAILROAD  TIME. 

250.  The  railroad  companies  have  divided  the  country 
into  four  time  belts,  extending  north  and  south.  All  places 
in  each  belt  take  the  time  of  the  meridian  which  passes 


186 


LONGITUDE    AND    TIME. 


through,  or  near  the  middle  of  the  belt.     The  belts  are  as 
follows :  Eastern,  Central,  Mountain,  and  Pacific. 

The  standard  meridian  for  the  Eastern  belt  is  the  75th, 
for  the  Central  belt  the  90th,  for  the  Mountain  belt  the 
105th,  and  for  the  Pacific  belt  the  120th. 

These  standard  meridians  are  15  degrees  apart.  There- 
fore, when  it  is  noon  in  the  Eastern  belt,  it  is  11  a.m.  in  the 
Central  belt,  10  a.m.  in  the  Mountain  belt,  and  9  a.m.  in  the 
Pacific  belt. 


In  going  westward  into  another  time  belt,  the  traveller 
sets  his  watch  back  one  hour. 

In  travelling  eastward,  he  sets  his  watch  ahead  one  hour. 

When  it  is  noon  on  the  standard  meridian  of  each  belt,  it 
is  called  noon  at  all  places  in  the  belt. 

Note,  —  Time  reckoned  by  this  method  is  not  true  solar  time,  but 
it  secures  a  uniformity  of  time  which  is  very  desirable. 


The  time  in  general  use  is  Kailroad  or  Standard  time. 


WRITTEN   EXERCISES.  187 

Oral. 

5.  When  it  is  5  p.m.  Mountain  time,  what  is  the  time  in 
the  Pacific  belt  ? 

6.  When  it  is  11  a.m.  Pacific  time,  what  is  the  Central 
time? 

7.  In  travelling  from  San  Francisco  to  New  York,  how 
many  times  do  I  change  my  watch,  and  do  I  set  it  ahead 
or  back  ? 

8.  When  it  is  4  a.m.  at  Augusta,  Me.,  what  is  the  stand- 
ard time  at  St.  Louis  ? 

9.  When  it  is  1  p.m.  Mountain  time  at  Denver,  what 
time  is  it  at  Washington,  D.C.  ? 

10.  What  is  the  Pacific  time  at  San  Francisco  when  it  is 
5  P.M.  at  Chicago  ? 

Written. 

11.  The  longitude  of  St.  Paul  is  93°  4'  55"  west,  of 
Philadelphia  is  75°  10'  west.  What  is  the  difference  in 
longitude  ? 

12.  The  longitude  of  New  York  is  74°  3"  west,  of  Paris 
is  2°  20'  12"  east.     What  is  the  difference  in  longitude  ? 

13.  New  York  City  is  74°  4"  west  from  London.  When 
it  is  noon  at  London,  what  is  the  true  time  at  New  York  ? 

14.  The  longitude  of  Boston  is  71°  4'  west,  and  Chicago 
is  87°  36'  west.  Chicago  is  how  far  due  west  from  Boston, 
if  there  are  51.27  miles  in  one  degree  at  their  latitude  ? 

15.  A  person  travelled  until  his  watch  was  3  hours  too 
fast.     In  what  direction  and  how  far  did  he  go  ? 

16.  What  is  the  difference  in  standard  time  between 
Boston  and  Chicago? 

17.  If  a  person  goes  from  New  York  to  San  Francisco, 
will  his  watch  be  too  fast  or  to  slow,  and  how  much  ? 

18.  The  difference  in  longitude  between  two  places  is 
17°  54'  55".     What  is  the  difference  in  time  ? 


188  LONGITUDE  AND  TIME. 

19.  The  longitude  of  San  Francisco  is  122°  26'  15"  west, 
and  that  of  Cincinnati  is  84°  26'  west.  When  it  is  9  a.m.  at 
San  Francisco,  what  is  the  time  at  Cincinnati  ? 

20.  The  longitude  of  Boston  is  71°  3'  30"  west,  and  that 
of  Paris  is  2°  20'  12"  east.  When  it  is  30  min.  past  2  p.m. 
at  Paris,  what  is  the  time  at  Boston  ? 

21.  Chicago  is  87°  38'  west.  When  it  is  27  min.  36  sec. 
past  11  A.M.  at  Chicago,  it  is  10  min.  past  12  m.  at  Washing- 
ton.    What  is  the  longitude  of  Washington  ? 

22.  St.  Louis  is  90°  15'  15"  west  longitude.  A  gentleman 
arriving  there  from  Boston,  71°  3'  SO"  west,  finds  that  his 
watch,  which  was  set  at  Boston,  is  not  right.  What  change 
must  he  make  ? 

23.  Mr.  Jones  started  from  Philadelphia,  and  travelled 
until  his  watch  was  1  hour  30  min.  slow.  How  many- 
degrees  did  he  travel,  and  in  what  direction  ? 

24.  When  it  is  12  o'clock  noon  at  Chicago,  what  time  is 
it  in  a  place  60°  30'  30"  west  of  Chicago  ? 

25.  Two  men  start  from  the  same  place,  and  travel  in 
the  same  direction,  one  going  3  degrees  and  the  other  5 
degrees  per  day.  They  travel  until  their  difference  in  time 
is  4  hours.     How  many  days  are  they  travelling  ? 

26.  What  is  the  difference  of  longitude  between  two 
places  when  their  difference  of  time  is  6  hr.  15  min.  12  sec.  ? 

27.  The  difference  in  time  between  two  places  is  8  hr. 
What  is  the  difference  in  longitude  ? 

28.  The  longitude  of  New  York  is  74°  3"  west.  When  it 
is  6  P.M.  at  New  York  it  is  49  min.  35  sec.  past  11  p.m.  at 
Berlin.     What  is  the  longitude  of  Berlin  ? 

29.  The  longitude  of  Philadelphia  is  75°  10'  west.  When 
it  is  10  A.M.  at  New  York  it  is  50  min.  55  sec.  past  6  a.m.  at 
San  Francisco.     What  is  the  longitude  of  San  Francisco  ? 


EEVIEW.  189 

REVIEW  OF  DENOMINATE  NUMBERS. 

251.  1.  Define  a  simple  number;  denominate  number; 
compound  number. 

2.  For  what  is  linear  measure  used  ? 

3.  For  what  is  square  measure  used  ? 

4.  For  what  is  cubic  measure  used  ?     !-  Give  tables. 

5.  For  what  is  liquid  measure  used  ? 

6.  For  what  is  dry  measure  used  ? 

7.  For  what  is  Troy  weight  used?  Give  the  table. 
Avoirdupois  weight  ?  Give  the  table.  Apothecaries' 
weight?      Give  the  table. 

8.  How  many  grains  in  a  pound  Troy  ?     Avoirdupois  ? 

9.  How  many  grains  in  an  ounce  Troy ?     Avoirdupois? 

10.  What  is  a  long  ton,  and  how  is  it  used  ? 

11.  How  many  days  in  a  common  year?  a  leap  year? 
What  is  the  solar  year  ?  Explain  leap  year.  When  does 
the  civil  day  begin  and  end  ? 

12.  What  is  the  use  of  circular  measure  ?  Define  circle, 
circumference,  diameter,  radius,  arc.  Give  the  table.  What 
is  the  measure  of  an  angle  ?  What  is  a  degree  ?  A  quad- 
rant ?  How  do  we  find  circumference  ?  How  do  we  find 
diameter  ?     What  is  a  right  angle  ? 

13.  What  is  a  surface?  a  square?  a  rectangle?  a  tri- 
angle ? 

14.  Give  the  rule  to  find  the  area  of  a  square;  of  a 
rectangle  ;  of  a  triangle  ;  of  a  circle. 

15.  Define  solid,  rectangular  solid,  cube,  cylinder.  How 
do  we  find  the  volume  of  a  rectangular  solid  ?  Of  a  cylinder  ? 

16.  What  is  reduction  ?  Reduction  ascending  ?  Eeduc- 
tion  descending  ? 


190  DENOMINATE   NUMBERS. 

17.  Define  a  denominate  fraction.  Give  the  different 
kinds  of  reduction  of  denominate  fractions. 

18.  How  do  we  add  •  compound  numbers  ?  subtract  ? 
multiply  ?  divide  ? 

19.  Give  the  common  method  of  finding  the  difference 
between  dates.     How  do  we  find  the  exact  difference  ? 

20.  What  is  longitude?  How  do  we  find  difference  in 
longitude  between  two  places  on  the  same  side  of  a  prime 
meridian  ?     On  opposite  sides  ? 

21.  How  do  we  find  difference  in  longitude  when  differ- 
ence of  time  is  given?  How  do  we  find  difference-of  time 
when  difference  in  longitude  is  given  ? 

22.-  What  is  standard  time?  What  are  the  names  of 
the  four  time  belts  ?  In  passing  west  into  a  time  belt, 
how  does  the  traveller  set  his  watch  ?     In  travelling  east  ? 

23.  How  do  we  find  length  when  area  and  breadth  are 
given  ? 

24.  How  do  we  find  length  when  volume,  thickness,  and 
width  are  given  ? 

25.  What  are  the  dimensions  of  a  rectangular  solid? 

26.  Define  cancellation,  even  number,  odd  number,  prime 
number,  composite  number. 

27.  When  are  numbers  prime  to  each  other  ? 

28.  How  many  cubic  feet  in  a  cord  of  wood  or  stone  ? 
How  long,  wide,  and  high  is  a  cord  of  wood  ?  What  is  a 
cord  foot  ? 

29.  For  what  is  board  measure  used?  What  is  a  board 
foot  ?     Give  the  rule  for  finding  board  feet. 

30.  How  do  we  find  the  capacity  of  bins  ?     Of  cisterns  ? 


THE    METRIC    SYSTEM 


LINEAR  MEASURE. 

252.  The  standard  unit  of  Linear  Measure  in  the  Metric 
System  is  the  Meter.  It  is  determined  by  taking  one  ten- 
millionth  part  of  the  distance  from  the  earth's  equator  to 
either  of  its  poles,  measured  on  a  meridian.  It  is  equal  to 
39.37  inches. 

QUESTIONS. 

253.  1.  What  denomination  in  the  English  linear  meas- 
ure is  most  nearly  like  the  meter  ? 

2.  Draw  a  line  one  meter  long. 

3.  Hold  your  hands  one  meter  apart. 

4.  A  meter  is  about  how  many  feet  long  ? 

5.  How  many  meters  long  is  your  schoolroom?  Wide? 
High? 

6.  About  how  many  meters  in  a  rod  ? 

HOW  THE  TABLE  IS  MADE. 

254.  Divide  a  meter  into  ten  equal  parts.  One  of  these 
parts  is  a  Decimeter.  Dec  is  a  Latin  stem  meaning  tenth. 
About  how  many  inches  long  is  a  decimeter?  Show  with 
your  hands  the  length  of  a  decimeter.  What  part  of  a 
meter  is  a  decimeter? 

191 


192  METRIC    SYSTEM. 

255.  Divide  a  decimeter  into  ten  equal  parts.  One  of 
these  parts  is  a  Centimeter.  Cent  is  a  Latin  stem  meaning 
hundredth.  What  part  of  an  inch  is  a  centimeter  ?  Show 
its  length.  How  many  centimeters  in  one  meter  ?  What 
part  of  a  meter  is  a  centimeter  ? 

256.  Divide  a  centimeter  into  ten  equal  parts.  One  of 
these  parts  is  a  Millimeter.  Mill  is  a  Latin  stem  meaning 
thousandth.  What  part  of  a  meter  is  a  millimeter?  How 
many  millimeters  in  a  meter?  What  part  of  an  inch  is  a 
millimeter  ? 

257.  Ten  meters  make  one  Dekameter.  Deka  is  a  Greek 
stem  meaning  ten.  How  many  rods  in  a  dekameter? 
How  many  feet?  How  many  dekameters  long  is  your 
schoolroom  ? 

258.  Ten  dekameters  make  one  Hektometer.  Hekto  is  a 
Greek  stem  meaning  hundred.  How  many  meters  in  one 
hektometer  ?     How  many  feet  long  is  a  hektometer  ? 

259.  Ten  hektometers  make  one  Kilometer.  Kilo  is  a 
Greek  stem  meaning  thousand.  How  many  meters  in  one 
kilometer  ?     How  many  feet  ?     What  part  of  a  mile  ? 

260.  Ten  kilometers  make  one  Myriameter.  Myria  is  a 
Greek  stem  meaning  ten-thousand.  How  many  meters  in 
one  myriameter  ?     How  many  feet  ?     How  many  miles  ? 

261.  These  statements  may  be  combined  in  the  following 
table : 

10  Millimeters  (mm.)=  1  Centimeter  (cm.)    =      .3937  -f  in. 

10  Centimeters  =  1  Decimeter  (dm.)     =    3.937  +  in. 

10  Decimeters  =  1  Meter  (m.)  =  39.37  +  in. 

10  Meters  =  1  Dekameter  (Dm.)  =  32.808  +  ft. 

10  Dekameters  =  1  Hektometer  (Hm.)  =  19.927  +  rd. 

10  Hektometers  =  1  Kilometer  (Km.)    =      .621  +  mi. 

10  Kilometers  =  1  Myriameter  (Mm.)=    6.213  +  mi. 


REDUCTION.  193 


262. 

1  Myriameter  = 

10  Kilometers  = 

100  Hektorneters  = 

1000  Dekameters  = 

10000  Meters  = 

100000  Decimeters  = 

1000000  Centimeters  = 

10000000  Millimeters. 


BEDDOTION. 


1  Millimeter  = 

.1  Centimeter  = 

I                       .01  Decimeter  = 

g                     .001  Meter  = 

g                   .0001  Dekameter  = 
J                 .00001  Hektometer  = 

.000001  Kilometer  = 


^1 


.0000001  Myriameter. 


263.  The  following  series  of  numbers  read  from  the  top 
downward  is  reduction  ascending;  read  from  the  bottom 
upward  is  reduction  descending.  All  metric  numbers  may 
be  reduced  in  this  way. 


75689132.  mm.  = 
7568913.2  cm.  = 
756891.32  dm.  = 
75689.132  m.  = 
7568.9132  Dm.  = 
756.89132  Hm.  = 
75.689132  Km.  = 
7.5689132  Mm.  = 


s  a  s  a   .  s  d  s 


All  these  numbers  might  be  read  thus :       7568913  2. 

QUESTIONS. 

264.    1.    How  can  a  metric  number  be  reduced  to  higher 
denominations  ?     To  lower  ? 

2.  Eeduce   12345678   mm.    to   cm.;    to  dm.;    to  m.;   to 
Dm. ;  to  Hm, ;  to  Km. ;  to  Mm. 

3.  Eeduce  9.6538714  Mm.  to  Km.;  to  Hm. ;  to  Dm.;  to 
m. ;  to  dm. ;  to  cm. ;  to  mm. 

4.  Reduce  7  Mm.  to  lower  denominations. 

5.  Reduce  7  mm.  to  higher  denominations. 


194  METRIC    SYSTEM. 

6.  Keduce  6307.1  m.  to  Km. ;  to  cm. 

7.  Reduce  31  meters  to  inches. 

8.  Write  2  Mm.  as  meters;  7  Km.;  6  Hm.;  8  Dm.;  5  m. 
3  dm. ;  2  cm. ;  9  mm.     Write  them  all  as  one  number. 

9.  Reduce  1  Mm.  to  feet. 

10.  Write  7  Mm.  and  6  mm.  in  one  number,  as  meters. 
Reduce  it  to  higher  denominations ;  to  lower. 

11.  Reduce  .075  Km.  to  cm. 

12.  Reduce  8  Dm.  and  6  m.  to  Mm. ;  to  mm. 

13.  Write  75  Km.  and  62  dm.  in  one  number  as  meters ; 
as  cm. ;  as  Mm. 

14.  State  the  value  of  each  figure  in  30769.543  m. 

15.  A  ship  sails  100  Mm.  in  one  day.      How  many  miles 
does  it  sail  ? 

16.  Give  the  table  of  Metric  Linear  Measure. 

17.  Name  the  standard  unit. 

18.  How  is  it  determined  ? 

19.  What  is  the  scale  of  the  Metric  system  ? 

20.  Name  in  order  the  Latin  and  Greek  stems  used  in 
the  table. 

SURFACE    MEASURE. 

265.  The  standard  unit  of   surface  measure  is  the  Are 
(pronounced  like  the  English  air). 

The  Are  is  a  square  whose  side  is  one  dekameter.     It  is 
therefore  a  Square  Dekameter. 

QUESTIONS.    ' 

266.  1.    An  are  is  how  many  meters  long  ?     Wide  ? 

2.  How  many  square  meters  does  the  are  contain? 

3.  An  are  is  how  many  inches  long  ?     Feet  ? 

4.  The  are  is  about  how  many  rods  long  ? 


SURFACE   MEASURE.  195 

5.  About  how  many  square  rods  does  it  contain  ? 

6.  About  how  many  ares  equal  one  acre  ? 

7.  How  many  ares  does  the   floor   of  your   schoolroom 
contain  ? 

8.  Name  all  the  surfaces  you  can  think  of  that  contain 
about  one  are. 


267.  The  table  of  surface  measure,  like  that  of  linear 
measure,  is  made  by  prefixing  the  Latin  and  Greek  stems 
to  the  standard  unit,  thus  : 

10  Centares  (ca.)  =  1  Deciare,  da. 
10  Deciares  =  1  Are,  a. 

10  Ares  ^  =  1  Dekare,    Da. 

10  Dekares  =  1  Hektare,  Ha. 

Note. — The  denominations  of  the  above  table  are  little  used, 
except  the  are,  the  hektare,  and  the  centare,  which  are  employed 
chiefly  in  measurements  of  land. 

268.  Draw  a  square  whose  side  is  one  meter.  How 
many  square  meters  does  it  contain  ?  It  is  how  many 
decimeters  on  a  side  ?  How  many  square  decimeters  does 
it  contain  ?  How  many  square  decimeters  make  one  square 
meter  ? 

269.  Draw  a  square  whose  side  is  one  decimeter.  How 
many  square  decimeters  does  it  contain  ?  How  many  centi- 
meters long  and  wide  is  it  ?  How  many  square  centimeters 
does  it  contain  ?  How  many  square  centimeters  in  one 
square  decimeter  ?  In  the  same  way  find  how  many  square 
millimeters  in  one  square  decimeter. 

How  many  sq.  Meters  =  1  sq.  Dekameter  ? 

How  many  sq.  Dekameter s    =  1  sq.  Hektometer  ? 
How  many  sq.  Hektometers  =  1  sq.  Kilometer  ? 


196  METRIC   SYSTEM. 

270.  The  answers  to  the  above  questions  form  the  follow- 
ing table  of  surface  measure,  which  is  used  for  all  ordinary 
surface  measurements : 

100  sq.  Millimeters  (sq.  mm.)  =  I  sq.  Centimeter,    sq.  cm. 
100  sq.  Centimeters  =  1  sq.  Decimeter,     sq.  dm. 

100  sq.  Decimeters  =  1  sq.  Meter,  sq.  m. 

100  sq.  Meters  =  1  sq.  Dekameter,    sq.  Dm. 

100  sq.  Dekameters  =  1  sq.  Hektometer,  sq.  Hm. 

100  sq.  Hektometers  =  1  sq.  Kilometer,  '  sq.  Km. 

QUESTIONS. 

271.  1.  Which  denomination  of  this  table  is  like  the  are  ? 

2.  Like  the  centare  ? 

3.  Like  the  hectare  ? 

4.  How  far  to  the  right  must  the  decimal  point  be  moved 
to  reduce  sq.  m.  to  sq.  dm.  ? 

5.  How  many  places  to  the  left  must  the  decimal  point 
be  moved  to  reduce  sq.  m.  to  sq.  Dm.  ? 

6.  To  reduce  sq.  mm.  to  sq.  cm.  ? 

7.  To  reduce  sq.  mm.  to  sq.  dm.  ? 

8.  Eeduce  5555  ca.  to  Ha. 

9.  Eeduce  3333  Ha.  to  ca. 

10.  A  field  134  m.  long  and  7  Dm.  wide  contains  how 
many  sq,  m.  of  land  ? 

11.  How  many  ares  ? 

12.  How  many  Ha.  ? 

13.  How  many  sq.  Dm.  ? 

14.  How  many  sq.  Hm.  ? 

15.  How  many  sq.  cm.  ? 

16.  How  many  sq.  cm.  in  an  oblong  643  cm.  long  and 
2.5,m.  wide  ? 


VOLUME  MEASURE.  197 

17.  How  many  sq.  mm.  ? 

18.  How  many  sq.  Km.  ? 

19.  One  hectare  equals  about  how  many  acres  ? 

VOLUME    MEASURE. 

272.  The  unit  chiefly  used  in  measuring  wood  and  stone 
is  the  Stere  (pronounced  stair),  which  is  a  cube  whose  edge 
is  one  meter.  What  denomination  in  the  English  volume 
measure  is  most  nearly  like  the  stere  ?  How  many  cubic 
meters  does  the  stere  contain  ?  How  many  decisteres  ? 
How  many  centisteres  ?     How  many  millisteres  ? 

QUESTIONS. 

273.  A  cube  whose  edge  is  one  meter  long  contains  how 
many  cubic  meters  ?  It  is  how  many  dm.  long  ?  Wide  ? 
High?  How  many  cu.  dm.  does  it  contain?  How  many 
cu.  dm.  =  1  cu.  m.  ?  A  cube  whose  edge  is  1  dm.  contains 
how  many  cu.  dm.  ?  How  many  cm.  long  is  it  ?  Wide  ? 
High  ?  How  many  cu.  cm.  does  it  contain  ?  How  many 
cu.  cm.  =  1  cu.  dm.  ?  A  cube  whose  edge  is  1  cm.  contains 
how  many  cu.  cm.  ?  How  many  mm.  long  is  it  ?  Wide  ? 
High  ?  How  many  cu.  mm.  does  it  contain  ?  How  many 
cu.  mm.  =  1  cu.  cm.  ? 

274.  From  the  answers  to  the  above  questions  make  the 
following : 

TABLE  OF  VOLUME  MEASURE. 

1000  cu.  Millimeters  (cu.  mm.)  =  1  cu.  Centimeter,  cu.  cm. 
1000  cu.  Centimeters  =  1  cu  Decimeter,   cu.  dm. 

1000  cu.  Decimeters  =  1  cu.  Meter,  cu.  m. 

QUESTIONS. 

275.  1.  How  may  cubic  millimeters  be  reduced  to  cubic 
centimeters  ?     To  cubic  decimeters  ?     To  cubic  meters  ? 

2.    How  many  places  to  the  right  must  the  decimal  point 
be  moved  to  reduce  cubic  meters  to  cubic  millimeters  ? 


198  METRIC   SYSTEM. 

3.  Reduce  7  cu.  m.  to  cubic  millimeters. 

4.  Reduce  5  cu.  mm.  to  cubic  meters. 

5.  How  many  steres  in  one  cubic  meter  ? 

6.  A  pile  of  wood  is  30  dm.  long,  3  m.  wide,  and  18  dm. 
high.     How  many  cubic  meters  does  it  contain  ? 

7.  How  many  steres  ?     (Ex.  6.) 

8.  How  many  cubic  millimeters  ?     (Ex.  6.) 

9.  How  many  cubic  centimeters  of  air  in  an  empty  box 
2  m.  by  12  dm.  by  75  cm.  ? 

10.  How  many  cubic  decimeters  ?     (Ex.  9.) 

11.  How  many  steres  of  stone  in  a  wall  30  m.  long,  5  dm. 
thick,  and  250  cm.  high  ? 

CAPACITY  MEASURE. 

276.  The  metric  capacity  measure  takes  the  place  of  both 
the  liquid  and  the  dry  measure  of  the  English  system. 

The  standard  unit  of  capacity  measure  is  the  Liter  (pro- 
nounced leeter),  which  is  a  cube  whose  edge  is  one  decimeter. 

QUESTIONS. 

277.  1.    The  liter  is  what  part  of  a  meter  wide  ?     High  ? 
Long  ? 

2.  What  part  of  a  cubic  meter  does  it  contain  ? 

3.  About  how  many  inches  wide  is  it  ?    High  ?    Long  ? 
About  how  many  cubic  inches  does  it  contain  ? 

4.  Show  with  your  hands  how  wide,  high,  and  long  a 
liter  is. 

5.  What  denomination  of  English  dry  measure   corre- 
sponds most  nearly  to  the  liter? 

6.  Make  a  full-sized  picture  of  a  liter. 

7.  What  object  the  size  of  a  liter  do  you  know  ? 


CAPACITY  MEASURE.       •  199 


TABLE. 


278.    The  table  of  capacity  is  formed  similarly  to  the  other 
metric  tables,  and  is  as  follows : 

10  Milliliters  (ml.)  =  1  Centiliter,    cl. 


10  Centiliters 

=  1  Deciliter, 

dl. 

10  Deciliters 

=  1  Liter, 

1. 

10  Liters 

=  1  Dekaliter, 

Dl. 

10  Dekaliters 

=  1  Hectoliter, 

HI. 

10  Hektoliters 

=  1  Kiloliter, 

Kl. 

10  Kiloliters 

=  1  Myrialiter, 

QUESTIONS. 

Ml. 

279.    1.    How  many  liters  in  1  myrialiter?     In  1  ml.  ? 

2.  How  many  milliliters  in  1  Ml.  ? 

3.  Reduce  12345678  ml.  to  higher  denominations. 

4.  Read  the  above  number,  giving  each  figure  the  name 
of  the  denomination  it  represents. 

5.  Reduce  154.67  cl.  to  Kl. 

6.  Reduce  .012346  Ml.  to  dl. 

7.  How  many  liters  equal  one  cubic  meter? 

8.  A  bin  is  2.5  m.  wide,  6.4  m.  long,  and  17  dm.  deep. 
How  many  liters  of  oats  will  it  hold  ?  How  many  HI.  ? 
How  many  Kl.  ? 

9.  A  tank  is  3  m.  long  and  3  m.  wide.  How  many  dm. 
deep  must  it  be  to  hold  50  HI.  of  water  ? 

10.    A  stone  containing  1  stere,  if  dropped  in  a  pond, 
would  displace  how  many  liters  of  water  ? 


200  METRIC   SYSTEM. 


MEASURES  OP   WEIGHT. 


280.  The  Gram  is  the  unit  of  weight.  It  is  equal  to  the 
weight  of  a  cubic  centimeter  of  distilled  water  at  its  greatest 
density. 

TABLE. 

10  Milligrams  (mg.)  =  1  Centigram,      eg. 


10  Centigrams 

=  1  Decigram,, 

dg. 

10  Decigrams 

=  1  Gram, 

g- 

10  Grams 

=  1  Dekagram, 

Dg. 

10  Dekagrams 

=  1  Hektogram, 

Hg. 

10  Hektograms 

=  1  Kilogram, 

Kg. 

10  Kilograms 

=  1  Myriagram, 

Mg. 

10  Myriagrams 

=  1  Quintal, 

Q. 

10  Quintals* 

=  1  Tonneau, 

T. 

or  Metric  Ton. 

The  weight  of  1  gram  is  15.432  grains. 

QUESTIONS. 

281.    1.    How  many  grams  in  1  metric  ton  ? 

2.  How  many  mg.  in  1  metric  ton  ? 

3.  Eeduce  1  mg.  to  T. 

4.  Reduce  1  T.  to  mg. 

5.  Reduce  9876543215  mg.  to  higher  denominations. 

6.  Read  the  above  number,  giving  each  figure  the  name 
of  the  denomination  it  represents. 

7.  Recite  the  table  of  weight. 

8.  Spell  the  name  of  each  denomination 

9.  Reduce  7.42  quintals  to  centigrams. 
10.  Reduce  543  mg.  to  Mg. 


MEASURES   OF   WEIGHT.  201 

11.  One  gram  equals  15.432  grains.     How  many  grains 
in  1  Kg.  ? 

12.  One    pound   Avoirdupois   contains   7000   gr.       How 
many  pounds  are  equivalent  to  one  Kg.  ? 

13.  Mr.  Smith  weighs  100  Kg.     How  many  pounds  does 
he  weigh  ? 

14.  How  many  grams  does  a  cubic  meter  of  distilled 
water  weigh  ? 

15.  Would  a  cubic  meter  of  any  other  substance  weigh 
the  same  as  a  cu.  m.  of  distilled  water  ?     State  your  reason. 

16.  How  many  kilograms  of  water  will  a  tank  4  m.  x 
3  m.  X  12  dm.  hold  ? 

REVIEW   QUESTIONS. 

282.    1.    How  many  tables  in  the  Metric  System  ? 

2.  Name  the  standard  units  in  the  order  in  which  they 
have  been  given.  Repeat  them  until  you  can  say  them  as 
rapidly  as  you  can  talk. 

3.  Name  the  prefixes  in  the  same  way. 

4.  Name  and  describe  the  unit  of  capacity  measure;  of 
weight ;  of  length ;  of  volume ;  of  surface. 

5.  Repeat  the  tables. 

6.  The  stere  is  the  unit  of  what  measure  ?  The  meter  ? 
The  are  ?     The  gram  ?     The  liter  ? 

7.  How  can  metric  numbers  be  reduced  to  higher 
'denominations  ?  to  lower  ? 

8.  How  many  things  are  to  be  committed  to  memory  in 
the  Metric  System  ? 

9.  What  is  39.37?  15.432?  10?  These  are  the  only 
numbers  that  need  be  remembered. 


GENEEAL   REVIEW. 


283.  1.  Define  fraction;  numerator;  fractional  unit; 
terms;    reduction  of  fractions. 

2.  Change  217  to  20ths. 

3.  Give  the  principle  upon  which  reduction  of  fractions 
is  based.     Illustrate. 

4.  Add  25^,  14|,  7|. 

5.  Give  the  rule  for  reducing  fractions  to  their  least 
common  denominator. 

6.  A  man  owned  J  of  a  foundry  and  sold  ^  of  his  share 
for  $  1200.     What  was  the  foundry  worth  ? 

7.  Eeduce  to  simple  form  (15f  -  3J)  x  (2i  +  5f). 

'•      6  +  8J        • 

9.  Reduce  to  least  common  denominator  six  thirty- 
fifths,  nine  twentieths,  and  five  sixteenths,  and  arrange  the 
results  according  to  value. 

10.  A  man  having  $  130  used  |  of  it.  How  much  of  it 
remained  ? 

11.  C  and  D  can  do  a  piece  of  work  in  24  days,  D  can  do 
it  alone  in  45  days.  How  many  days  will  C  require  to 
do  it? 

12.  The  numerator  of  a  fraction  is  6510,  the  denominator 
66495.     lleduce  the  fraction  to  its  lowest  terms. 

202 


"WRITTEN  EXERCISES.  203 

13.  If  7  be  added  to  each  term  of  the  fraction  f,  will  its 
value  be  increased  or  diminished,  and  how  much  ? 

14.  Two  men  are  140  miles  apart,  and  travel  towards 
each  other,  one  at  the  rate  of  3J  miles  an  hour,  and  the 
other  at  the  rate  of  4|^  miles  an  hour.  In  how  many  hours 
will  they  meet  ? 

15.  Define  decimal  fraction;  an  account;  currency. 

16.  What  will  6827  feet  of  lumber  cost  at  $10.50 
per  M.  ? 

17.  8.7625  +  31.735-17.382569  =  ? 

18.  AVrite  in  words  365.     8752. 

19.  Find  the  cost  of  7896  pounds  of  hay  at  f  16  a  ton. 

20.  Express  in  figures  two  hundred  sixty-five  and  five 
thousand  one  hundred  ten  millionths. 

21.  Change  .875  to  a  common  fraction  in  its  lowest  terms. 

22.  How  is  a  bill  receipted? 

23.  Give  a  rule  for  dividing  a  decimal  by  10,  100,  1000, 
etc. 

24.  Eeduce  3.25,  12.364,  and  .56087  to  a  common 
denominator. 

25.  When  will  a  fraction  reduce  to  a  perfect  decimal  ? 

26.  7.6875  --  187.5  x  (5|-  +  2f )  =  ? 

27.  How  is  the  place  for  the  decimal  point  in  the  j^roduct 
determined  ? 

28.  Give  the  abbreviations  of  Creditor  and  Merchandise. 

29.  Name  the  seventh  decimal  order. 

30.  James  Harris,  of  Denver,  Col.,  sold  for  cash  to 
Preston  White,  on  Nov.  4,  1901,  42  lb.  of  sugar  at  10  cents  ; 
3  lb.  Y.  H.  tea  at  f  .60;  4  gal.  molasses  at  $  .75 ;  48  yd.  sheet- 
ing at  $  .14 ;  1  box  starch  46  cents,  and  8  doz.  eggs  at  $  .24. 
Make  the  bill  in  due  form. 


204  GENERAL   REVIEW. 

31.  Define  a  square;  a  circle. 

32.  Write  the  table  of  cubic  measure. 

33.  For  what  purposes  are  the  following  used:  Troy 
weight  ?    Dry  measure  ? 

34.  The  last  war  with  England  commenced  June  18, 
1812,  and  ended  Feb.  17,  1815.     How  long  did  it  continue  ? 

35.  A  jeweller  made  3  lb.  2  pwt.  2  gr.  of  gold  into  rings 
weighing  5  pwt.  10  gr.  each.  How  many  rings  were 
there  ? 

36.  Reduce  2  mi.  6  ch.  3  rd.  to  links. 

37.  Eeduce  to  integers  of  lower  denominations  £|,  and 
.25256  T. 

38.  Change  4  S  5  3  2  3  8  gr.  to  a  decimal  of  a  pound. 

39.  Find  the  result  of  4.8  bu.  +  2|  bu.  +  .8125  pk.  -f- 
2f  pk.  +  }  bu. 

40.  A  grocer  bought  35  casks  of  molasses,  each  contain- 
ing 44  gal.  2  qt.  1  pt.     How  much  did  they  all  contain  ? 

41.  A  ship  in  8°  north  latitude  sailed  due  south  until  it 
reached  12°  south  latitude;  find  the  distance  it  sailed  in 
statute  miles. 

42.  Find  the  value  of  -^-^  of  a  ton. 

43.  Reduce  -^  of  a  year  to  integers  of  lower  denomi- 
nations. 

44.  Reduce  f  of  a  lb.  Troy  to  integers  of  lower  denomi- 
nations. 

45.  Express  120  rd.  2  yd.  1  ft.  6.  in.  as  the  fraction  of 
a  mile. 

46.  Reduce  45  sq.  rd.  2  sq.  ft.  9  sq.  in.  to  the  fraction  of 
an  acre. 

47.  What  part  of  a  day  are  6  hr.  13  min.  20  sec.  ? 


WRITTEN  EXERCISES.  205 

48.  What  part  of  4  gal.  2  qt.  1  pt.  are  1  gal.  1  qt. 
Ipt.? 

49.  At  25  cents  an  ounce,  what  is  the  value  of  18  oz.  10 
pwt.  12  gr.  of  silver  ? 

50.  How  much  will  it  cost  to  fill  a  bin  with  corn  at  $.45 
a  bushel,  if  the  bin  is  10  ft.  square  on  the  bottom  and  4 
ft.  deep. 

51.  A  cistern  measures  inside  the  walls  8  by  6  by  9  ft., 
and  lacks  1^  ft.  of  being  full.  How  many  gallons  does  it 
hold  ? 

52.  How  many  busliels  will  a  box  hold  of  the  same 
dimensions  as  in  Ex.  51  ? 

53.  How  many  cords  of  wood  in  a  pile  of  wood  that  is 
twice  the  length,  height,  and  width  of  an  established  cord  ? 

54.  Find  the  total  weight  of  5  car-loads  of  coal,  weigh- 
ing respectively  14  T  18  cwt.  63  lb.,  17  T.  4  cwt.  85  lb., 
13  T.  19  cwt.  26  lb.,  15  T.  10  cwt.  43  lb.,  and  14  T.  7  cwt. 
90  1b. 

55.  How  many  bricks  8  in.  by  4  in.  by  2  in.  will  it  take 
to  pave  a  street  ^  mile  long  and  -^^  mile  wide,  laying  the 
bricks  on  the  longest  narrow  face  ?  How  many  if  they  are 
placed  on  end  ? 

56.  How  much  wood  in  three  piles,  the  first  of  which 
contains  10  cd.  6  cd.  ft.  4  cu.  ft.,  the  second  12  cd.  12  cu. 
ft.,  the  third  17  cd.  1  cd.  ft.  ? 

57.  A  family  consumes  daily  6  lb.  14  oz.  of  bread.  If 
each  loaf  weighs  1  lb.  6  oz.  and  costs  7  cents,  how  much 
does  bread  cost  the  family  for  the  month  of  August  ? 

58.  Find  the  sum  of  f  mi.,  -|  fur.,  ^  rd.,  and  f  ft. 

59.  From  6^  mi.  take  4  mi.  140  rd.  4  yd. 

60.  A  man  has  a  bin  6  ft.  long,  4  ft.  wide,  3  ft.  deep, 

1  filled  with  wheat.     If  he  sells  10  sacks,  each  containing 

2  bu.  1  pk.  5  qt.,  how  much  is  left  ? 


206  GENERAL  REVIEW. 

61.  From  a  piece  of  land  20  rods  long,  180  ft.  wide,  were 
sold  4  lots,  each  50  ft.  wide,  150  ft.  long.  What  part 
remained  ? 

62.  A  merchant  bought  two  casks  of  wine,  each  contain- 
ing 41  gal.  3  qt.,  at  $1.80  per  gallon.  One-seventh  of  it 
leaked  away.  He  sold  9  kegs,  each  containing  5  gal.  1  qt., 
at  30^  a  pint,  and  the  remainder  at  40/  a  pint.  How 
much  did  he  gain  ? 

63.  A  coal-dealer  bought  34,160  lb.  of  coal  at  $2.50  per 
long  ton.  He  sold  8  loads,  each  1  T.  4  cwt.  60  lb.,  at  $  3 
per  ton,  and  the  rest  for  $3.25  per  ton.  How  much  did 
he  gain? 

64.  Change  3,895,504"  to  higher  denominations. 

65.  Three  quadrants  of  a  circle  are  equal  to  how  many- 
seconds  ? 

66.  Through  how  many  degrees  does  the  minute-hand 
of  a  clock  pass  in  2i  hours  ?  Through  how  many  does  the 
hour-hand  pass  in  the  same  time  ? 

67.  How  many  minutes  elapse  between  four  o'clock 
Friday  afternoon  and  nine  o'clock  the  following  Monday 
morning  ? 

68.  A  boy  was  exactly  10  years  old  when  the  United 
States  declared  war  against  Mexico,  May  13,  1846.  How 
old  was  he  at  the  time  of  the  first  bloodshed  of  the  Civil 
War,  April  19,  1861? 

69.  A  train  leaves  New  York  at  six  o'clock  Monday 
evening,  and  travels  an  average  of  J  of  a  mile  a  minute. 
When  will  it  reach  Buffalo,  a  distance  of  410  miles  ? 

70.  A  cistern  that  holds  50  bushels  is  6  ft.  square.  How 
deep  is  it  ? 

71.  A  pile  of  wood  is  6  ft.  high  and  4  ft.  wide.  How 
long  must  it  be  to  contain  3  cords  ? 


WRITTEN    EXERCISES.  207 

72.  A  man  has  a  circular  garden  with  a  diameter  of 
36  feet.  How  many  rods  of  fencing  will  be  required  to 
enclose  it  ? 

73.  How  many  square  yards  in  the  above  garden?  (Ex.  72.) 

74.  A  city  lot  is  35  ft.  front  and  125  ft.  deep.  Find 
the  area. 

75.  A  meadow  contains  8|  acres.  Its  width  is  35  rods. 
Find  the  length  of  it. 

76.  How  many  yards  of  carpeting  j  of  a  yd.  wide  will 
be  required  for  a  room  18  ft.  wide  and  20  ft.  long,  if  the 
strips  run  lengthwise,  and  there  is  a  waste  of  6  in.  in  each 
strip  for  matching  patterns  ? 

77.  The  platform  in  a  schoolroom  is  30  ft.  long  and  11 
ft.  wide.  What  will  be  the  cost  of  oil-cloth,  at  85  cents 
a  square  yard,  to  cover  it  ? 

78.  How  many  feet,  board  measure,  in  6  boards  16  ft. 
long,  10  in.  wide,  1  in.  thick  ? 

79.  Find  the  cost  of  10  Norway  sidewalk  planks  16  ft. 
long,  12  in.  wide,  2  in.  thick,  at  $18  per  M. 

80.  A  class-room  is  15  ft.  long,  12  ft.  wide,  10  ft.  high. 
Find  the  cost  of  plastering  it  at  20  cents  a  yard. 

81.  A  room  is  30  ft.  wide,  and  40  ft.  long,  and  16  ft.  high. 
Find  the  number  of  square  yards  of  plastering  in  it  after 
making  allowance  for  wainscoting  3  ft.  high,  8  windows, 
4  ft.  by  8  ft.,  and  6  doors,  3  ft.  6  in.  by  7  ft.  6  in. 

82.  What  will  it  cost  to  paper  a  kitchen  12  ft.  by  11  ft. 
and  9  ft.  high,  with  10-cent  paper,  if  each  roll  covers  4  sq. 
yd.? 

83.  Find  the  cost  of  papering  a  room  16  ft.  long,  12  ft. 
wide,  9  ft.  6  in.  high,  with  paper  18  in.  wide,  8  yd.  in  a 
roll,  at  50  cents  a  roll,  if  20  sq.  yd.  be  allowed  for  doors, 
windows,  and  base-boards. 


208  GENERAL   REVIEW. 

84.  If  a  shingle  is  4  in.  wide,  and  lies  5-|  in.  to  the 
weather,  how  many  shingles  will  it  take  to  shingle  one  side 
of  a  roof  that  is  32  ft.  long  by  22  ft.  wide,  allowing  an  extra 
course  at  the  eaves  ?     How  many  for  both  sides  ? 

85.  What  would  be  the  cost  of  the  shingles  for  both  sides 
of  the  roof  in  Ex.  84  at  $3.25  per  M.  ? 

86.  The  product  of  two  numbers  is  lyij- ;  one  of  the  num- 
bers is  |.     What  is  the  other  ? 

87.  What  fraction  multiplied  by  -f-  will  equal  -^  ? 

88.  How  many  square  yards  of  carpet  will  be  required 
to  carpet  a  room  that  is  27  ft.  by  33  ft.  ?  How  many  yards 
of  carpet  will  be  required  if  the  carpet  is  30  in.  wide  ? 

89.  If  a  hotel  uses  3  pounds  of  coffee  a  week,  what  would 
be  paid  for  coffee  at  38  cents  a  pound  for  January,  February, 
and  March,  1896  ? 

90.  When  it  is  noon  at  New  York,  73°  59'  9"  W.,  what 
is  the  time  at  Chicago,  87°  36'  42"  W.  ?  What  is  the  time 
at  New  York  when  it  is  noon  at  Chicago  ? 

91.  When  it  is  noon  at  Greenwich,  what  is  the  longitude 
of  a  place  whose  time  is  8.30  a.m.  ? 

92.  A  and  B  start  at  a  given  point,  and  travel  in  opposite 
directions.  A  travels  until  his  longitude  is  30°  40'  greater 
than  it  was,  and  B  travels  half  as  far  as  A.  What  is  the 
difference  in  time  between  the  places  they  are  then  in  ? 

93.  What  part  of  a  pound  Avoirdupois  is  a  pound  Troy  ? 

94.  What  part  of  an  ounce  Troy  is  an  ounce  Avoirdupois  ? 

95.  A  druggist  bought  opium  at  $  8  a  pound  Avoirdupois, 
and  sold  it  at  75^  an  ounce  Troy.  What  was  his  profit  on 
10  pounds  ? 

96.  How  much  heavier  is  a  pound  of  iron  than  a  pound 
of  gold  ? 


WRITTEN  EXEBCISES.  209 

97.  What  is  the  difference  in  the  areas  of  two  fields,  one 
being  5  Hm.  long  and  8  Dm.  wide,  the  other  8  Hm.  long 
and  14  Dm.  wide  ? 

98.  In  a  cubic  dekameter  how  many  cubic  millimeters  ? 

99.  A  rectangular  field  is  5.4  Hm.  long  and  1.5  Hm. 
wide.     How  many  hektares  does  it  contain  ? 

100.  Three  fields  have  an  area  respectively  of  19  A.  146 
sq.  rd.,  12  A.  73  sq.  rd.  15  sq.  yd.,  and  9  A.  127  sq.  rd.  26 
sq.  yd.     What  is  the  total  area  ? 

101.  Find  the  volume  and  the  area  of  the  curved  surface 
of  a  cylinder  whose  diameter  is  8  in.,  and  whose  altitude  is 
11  in. 

102.  How  many  square  feet  of  glass  in  6  windows  of  8 
panes  each,  each  pane  being  14  by  12  inches  ? 

103.  What  will  it  cost  to  pave  a  street  2^  miles  long  and 
2 J  rods  wide,  at  $  12  per  square  rod  ? 

104.  How  many  yards  of  oil  cloth  8  ft.  wide  will  cover  a 
floor  32  by  22  ft.,  if  the  strips  run  lengthwise,  allowing 
1  yard  for  waste  ? 

105.  How  many  yards  of  crash  18  inches  wide  will  cover 
a  floor  18  ft.  square,  allowing  If  yards  for  waste  ? 

106.  How  many  sq.  yards  to  be  plastered  in  the  ceiling 
and  walls  of  a  room  26  by  20  ft.  and  12  ft.  high,  there  being 
3  windows  6  ft.  by  2  ft.  8  in.,  and  2  doors  8  by  3  ft.  ? 

107.  How  many  square  feet  of  boards  in  a  tight  fence  10 
rods  long  and  6  ft.  high  ? 

108.  How  many  sq.  rods  in  a  garden  231  ft.  long  and 
165  ft.  wide? 

109.  What  will  it  cost,  at  25^  a  sq.  foot,  to  construct  a 
slate  blackboard  34  ft.  3  in.  long  and  4  ft.  6  in.  wide  ? 


PERCENTAGE, 


284.   Oral. 

How  much  is  J  of  20  ? 

5  is  ^  of  what  ? 

5  is  how  many  hundredths  of  20  ? 

Questions  of  Relation  may  be  solved  by  means  of  hun- 
dredths; thus, 

a.    How  much  is  ^%  of  20  ? 

6.    5  is  j\\  of  what  ? 

c.    5  is  how  many  hundredths  of  20  ? 

Another  name  for  hundredths  is  per  cent;  thus,  -f^-^  is 
25  per  cent,  -^i-o  =  ^  P^^  cent,  .16  =  16  per  cent,  .05  =  5 
per  cent. 

The  sign  of  per  cent  is  %.  25  per  cent  is  25%,  6  per 
cent  is  6%,  |  per  cent  is  ^%. 

Read  questions  a,  6,  and  c,  using  the  name  per  cent 
where  necessary. 

Write  questions  a,  b,  and  c,  using  the  sign  %  in  its 
proper  place. 

Solve  questions  a,  b,  and  c,  using  decimal  per  cent, 

210 


WRITTEN  EXERCISES.  211 

285.   Percentage  is  a  process  of  solving  questions  of  rela- 
tion by  means  of  hundredths. 

Written. 

1.  How  much  is  3%  of  400  ? 

Solve  the  above,  form  question  &,  and  solve  it. 

2.  How  much  is  10%  of  200  ? 

Solve  the  above,  form  question  c,  and  solve  it. 

Solve  the  following  questions,  then  form  questions  h  and 
c,  and  solve  them : 

3.  How  much  is  4  per  cent  of  $  200  ? 

4.  30%  of  500  is  how  much  ? 
6.    How  much  is  50%  of  90  ? 

6.  A  boy  earned  $4.00,  and  spent  10%  of  it  for  a  book. 
What  was  the  cost  of  the  book  ?     (Question  a.) 

7.  A   farmer   had   100  sheep  and  sold  20%   of   them. 
How  many  sheep  did  he  sell  ? 

Solve  the  following,  form   questions  a  and  c,  and  solve 
them  : 

8.  15  is  10%  of  what  number? 

9.  160  is  80%  of  what  number  ? 

10.  50  is  25%  of  what  number  ? 

11.  A  boy  lost  20  cents,  which  was  5  per  cent  of  all  his 
money.     How  much  money  did  he  have  ?     (Question  h.) 

12.  A    farmer   sold  150   sheep,   which  was   50%   of  his 
entire  flock.     How  many  sheep  were  in  the  flock  ? 

Solve  the  following,  form  questions  a  and  h,  and  solve 
them : 

13.  $  20  is  what  per  cent  of  $  100  ? 

14.  What  per  cent  of  60  is  15  ? 

15.  40^  is  what  %  of  $4.00  ? 


212  PERCENTAGE. 

16.  A  boy  earns  $  5.00  a  week,  and  saves  $  2.00  of  it. 
What  per  cent  of  his  money  does  he  save  ?     (Question  c.) 

17.  A  dealer  bought  a  gross  of  pencils,  and  sold  36  of 
them.  What  per  cent  of  his  pencils  did  he  sell  ?  What 
per  cent  remained  unsold  ? 

Change  the  following  fractions  to  others  having  100  for 
a  denominator:  i;  |;  /^;  |;  -^\',  J-^;  -i;  |;  -\;  ^\. 

Change  the  above  fractions  to  decimal  hundredths. 


Eead  as  hundredths :  .05;  .186;  .331;  .24^;  .27 J;  .2725; 

;  .1. 

Write  the  above  in  hundredths  as  common  fractions. 


.5;  .1. 


286.  E-ead  the  following  Questions  of  Relation : 
Question  a.     How  much  is  5%  of  200  ?     Ans.  10. 
Question  b.     10  is  5%  of  what  ?     Ans.  200. 
Question  c.     10  is  what  %  of  200  ?     Ans.  5%. 

These  three  kinds  of  questions  form  the  basis  of  a  great 
variety  of  practical  computations,  which  are  classed  under 
the  general  head  of  Percentage. 

287.  Every  question  in  percentage  involves  three  ele- 
ments :  the  Rate  per  cent,  the  Base,  and  the  Percentage. 

The  Rate  per  cent  is  the  number  of  hundredths  taken.  In 
question  a,  what  is  the  rate  per  cent  ? 

The  Base  is  the  number  of  which  the  hundredths  are 
taken.     In  question  a,  what  is  the  base  ? 

The  Percentage  is  the  result  obtained  by  taking  a  cer- 
tain per  cent  of  a  number.  In  question  a,  what  is  the 
percentage? 


,/wwex  -y^  Hi  'Vm^-^-^^-^aaw -ux#|,t^ 


WRITTEN   EXERCISES.  213 


How  much  is  8%  of  $200 


Solution.  —8%  of  $200  =  200  x  .08  =  $16.     We  now  have  the 
three  elements,  as  follows: 

8  %  is  the  rate,  $  200  is  the  base,  and  $  16  is  the  percentage.   ■ 

Since  $200  x  .08  =  $  16,  the  percentage ; 
$  16  -~  .08  =  §200,  the  base ; 
And  $  16  -^  -1200  =  .08,  the  rate. 

288.    Therefore,  when   any  two   of    these   elements   are 
given,  the  other  may  be  found,  thus: 

Base  X  Rate  =  Pernenta^e; 
Percentage  -^  Pasp  —  Pt^tft 


289.  Tell  which  elements  are  given,  and  which  one  is 
required,  in  question  a ;  in  question  h ;  in  question  c. 

290.  Find  the  percentage  and  form  questions  h  and  c, 
but  do  not  solve  them. 

18.  6%  of  100  is  what? 

19.  How  much  is  25%  of  200? 

20.  How  much  is  40%  of  250? 

21.  What  is  4%  of  50  men? 

22.  20%  of  80  is  what? 

23.  15%  of  f  40  =  ? 

24.  What  is  3%  of  400  gallons? 

25.  What  is  90%  of  200  pounds? 

26.  60%  of  200  miles  =  ? 

27.  10%  of  15  inches  =  ? 

28.  What  is  the  base  in  each  of  the  above  questions? 


214  PERCENTAGE. 

291.  Care  should  be  taken  to  express  the  decimal  rate 
per  cent  properly,  as  hundredths.  Every  fractional  part 
of  1%  must  be  written  at  the  right  of  the  hundredths 
place. 


1%  =  .01 

12^%  =  .12^  or  .125 

9%  =  .09 

J%  =  .00^  or  .005 

10%  =  .10 

10tL%  =  .107 

90%  =  .90 

33i%  =  .331 

100%  =  1.00 

81%  =  .08 J  or  .0825 

900%  =  9.00 

J%  =  .OOi  or  .0025 

125%  =  1.25 

^%  =  .001  or  .00125 

29 

2.   Express  decimally: 

1. 

7%               6. 

6i% 

11.    101%           16.    ^% 

2. 

6%                 7. 

12i% 

12.    110%            17.    f% 

3. 

2%                 8. 

15|% 

13.    250%            18.    -S-% 

4. 

12%               9. 

37i% 

14.    200%             19.     1% 

5. 

78%           10. 

H% 

15.    127|%          20.    ^^ 

293.    It  is  often  convenient  to  change  the  rate  per  cent 
to  the  common  fraction  form ;  thus  : 


100    100     2     ;^0    ^ 

4 


Change  to  common  fractions  in  lowest  terms : 

1.  25%    ■         5.    16f%              9.    150%           13.  f% 

2.  50%             6.    33^%            10.    225%           14.  |% 

3.  75%             7.   37i%            11.    175%           15.  f% 

4.  20%             8.    87^%            12.    236%           16.  i% 

What  per  cent  of  a  number  is  i  of  it?  i?  |?  J?  f  ? 

1    ?   _9    ?    1  ? 
31F  •     iTF  •     ^  • 


WRITTEN   EXERCISES.  215 

294.    Express    in    both    the    decimal    and    the    common 
fraction  form : 


1.   25% 

/6. 

6|% 

9. 

108% 

13. 

f% 

2.    60% 

6. 

6i% 

10. 

150% 

14. 

1% 

3.    18% 

7. 

^i% 

11. 

125% 

15. 

\% 

4.    1% 

s. 

66|% 

12. 

137^% 

16. 

tV/ 

295.  Per  cent  is  commonly  used  in  the  decimal  form, 
but  many  operations  may  be  much  shortened  by  using  the 
common  fraction  form. 

Solve,  using  first  the  decimal,  then  the  common  fraction 
form,  and  note  the  difference : 

17.  How  much  is  25%  of  $  324  ? 

18.  Find  12^%  of  960  sheep. 

19.  What  is  16|%  of  366  men? 

20.  Find  331%  of  12  oranges. 

21.  50%  of  4  tons  is  what  ? 

22.  20%  of  f  300  =  what? 

Question  a,  Oral. 

23.  What  is  yf  (^  of  800  ?  27.  50%  of  144  men  ? 

24.  What  is  y^^  of  900  ?  28.  20%  of  15  eggs? 
25.,  16|%  of  48  apples  ?  29.  .07  of  500  ? 

26.    33i%  of  m  sheep?  30.    12J%  of  $16? 

Written. 

296.  Rate  and  base  given,  to  find  percentage. 

Base  X  Rate  =  Percentage. 

1.  What  is  40%^  of  $  120-? 

2.  How  much  of  12^%  of  1600  lb.? 
8.    18j-\%  of  365  is  what? 


216  PERCENTAGE. 

4.  From  a  flock  of  60  sheep,  10%  were  sold.  How  many- 
were  sold?     The  question  is,  "How  much  is  10%  of  60?" 

5.  How  much  is  100%  of  50  bushels? 

6.  A  man  having  50  bushels  of  wheat  sold  20  per  cent 
of  it.     How  many  bushels  did  he  sell? 

7.  A  man  had  $  1500  in  the  bank  and  drew  out  40% 
of  it.     How  much  remained  in  the  bank  ? 

Note. —100%  represents  all  he  had  in  the  bank.  100% -40% 
=  60  %,  the  part  that  remained.  The  question  then  becomes,  How 
much  is  60%  of  $1500? 

8.  A  farmer  having  320  acres  of  land  sold  15%  of  it 
to  one  man  and  25%  to  another.  How  many  acres  did 
he  sell? 

9.  A  wholesale  grocer  had  480  bbl.  of  A  sugar,  and  sold 
12:^%  of  it.     How  much  remained  unsold? 

10.  How  much  is  .5%  of  80  ? 

11.  How  much  is  i%  of  $  4000  ? 

Question  b,  Oral. 

1.  5  is  ^25^  of  what  ?  4.  12  is  8%  of  what ? 

2.  5  is  25%  of  what  ?  5.  30  is  12^%  of  what  ? 

3.  40  is  10%  of  what?  6.  15  is  50%  of  what? 

Written. 

297.    Percentage  and  rate  given,  to  find  base. 

Percentage  -^  Kate  =  Base. 

7.  $125isl2|-%  of  what?    • 

8.  150  bu.  is  33^%  of  what? 

9.  240  is  120%  of  what? 

10.  $  1644  is  40%  of  what  ? 

11.  75is3J%  of  what? 


WRITTEN   EXERCISES.  •  217 

12.  289  is  50%  of  what? 

13.  25%  of  my  property  is  $5000.  What  is  the  value 
of  my  property  ?    The  question  is,  "  $  5000  is  25  %  of  what  ?  " 

14.  I  sold  a  horse  for  $S1,  which  was  90%  of  what  it 
cost  me.     What  did  the  horse  cost  me  ? 

Question  c,  Oral. 

1.  What  part  of  45  is  15  ? 

2.  What  per  cent  of  45  is  15  ? 

3.  What  per  cent  of  80  is  60  ? 

4.  What  per  cent  of  90  is  30  ? 

5.  $40  is  what  per  cent  of  $60?  ' 

6.  12  yd.  is  what  per  cent  of  36  yd.  ? 

7.  14  bu.  is  what  per  cent  of  56  bu.  ? 

8.  f  is  what  per  cent  of  |^  ? 

Written. 

298.  Base  and  percentage  given,  to  find  rate. 

Percentage  -i-  Base  =  Rate. 

9.  What  per  cent  of  $  240  is  $  80  ? 

10.  150  is  what  per  cent  of  900  ? 

11.  What  %  of  a  long  ton  is  a  short  ton  ? 

12.  What  %  of  5  days  is  6  hours  ? 
y^  13.    5  cwt.  is  what  %  of  3  tons? 

14.  $  28.16  is  what  %  of  $  7040  ? 

15.  What  per  cent  is  i  of  2i  ?     ^  of  f  ?     f  of  7^  ? 

16.  My  salary  is  $  1600  and  my  expenses  $  1200.  What 
%  of  my  salary  are  my  expenses  ?  The  question  is,  "  $  1200 
is  what  %  of  $  1600  ?  " 

299.  The  sum  of  the  base  and  percentage  is  called  the 
Amount. 


218  '  PERCENTAGE. 

300.  The  difference  between  the  base  and  percentage  is 
called  the  Difference. 

Find  the  amount  in  the  following : 

1.  How  much  is  10%  of  20  ?     Find  the  difference. 

2.  20  is  what  per  cent  of  itself  ? 

3.  If  20  is  increased  by  10%  of  itself,  the  amount  is 
22.     What  per  cent  of  20  is  22  ? 

Solution.  —  The  base  ...     20  is  100  %  of  20. 
The  percentage  .       2  is     10  %  of  20. 
Therefore  the  amount  .     22  is  110%  of  20.    Ans. 

4-    100%  +10%  =  ?     1  +  10%=? 

5.  If  20  is  diminished  by  10%  of  itself,  the  difference 
is  18.     What  per  cent  of  20  is  18  ? 

Solution. —The  base  .  .  .  20  is  100%  of  20. 
The  percentage  .  2  is  10  %  of  20. 
The  difference    .     18  is    90%  of  20.     Ans. 

6.  100%  -  10%  =  ?     1  -  10%  =  ? 

7.  What  number  increased  by  10%  of  itself  equals  220  ? 

Solution.  —  Since  220  is  10%  more  than  the  required  number, 
220  is  110%  of  the  required  number. 

The  amount,  220,  is  now  treated  as  the  percentage,  and  110%  as 
the  rate:  and  the  question  becomes,  220  is  110%  of  what  number? 
(Question  &.) 

220-1.10  =  200.     Ans. 

Amount  -^-  (1  +  rate)  =  Base. 

8.  What  number  diminished  by  10%  of  itself  equals  180  ? 

Solution.  —  Since  180  is  10  %  less  than  the  required  number, 
180  is  90%  of  the  required  number. 

The  difference,  180,  is  now  treated  as  the  percentage,  and  90  %  as 
the  rate;  and  the  question  becomes,  180  is  90%  of  what  number? 
(Question  ft. ) 

180  -  .90  =  200.     Ans. 

Difference  -h  (1  —  rate)  =  Base. 


WRITTEN   EXERCISES.  219 

9.    What  number  increased  by  25%  of  itself  equals  290  ? 

10.  What  number  diminished  by  25%  of  itself  equals 
243  ? 

11.  After  selling  20%  of  his  sheep,  a  farmer  had  400 
sheep  left.     How  many  had  he  at  first? 

12.  The  population  of  a  certain  city  has  increased  12% 
in  two  years.  If  it  now  numbers  56000,  what  was  it  at 
the  beginning  of  the  two  years  ? 

13.  A  clerk's  salary  was  increased  6J%.  If  he  now 
receives  $  850,  what  was  his  original  salary  ? 

14.  By  selling  goods  at  $630  I  lose  121%.  What  did 
I  pay  for  them  ? 

15.  $580  is  10%  less  than  what  number? 

16.  I  sold  goods  at  $450,  which  was  120%  of  the  cost. 
What  was  the  cost? 

17.  After  withdrawing  45%  of  my  money  from  the  bank, 
I  still  have  $1300  on  deposit.  How  much  had  I  in  the 
bank  at  first. 

18.  A  farmer  increased  his  flock  of  sheep  by  12J%,  and 
then  had  900.     How  many  had  he  at  first  ? 

19.  A  man,  after  spending  a  month  in  the  Adirondacks, 
finds  that  his  weight  is  210  pounds,  which  is  an  increase  of 
5%.  What  was  his  weight  before  he  went  to  the  Adiron- 
dacks ? 

20.  A  regiment  lost  12|%  of  its  men  in  an  engagement, 
and  had  560  left.  How  many  men  were  there  before  the 
engagement  ? 

21.  A  owes  C  33J%  more  than  he  owes  B.  If  he  owes 
C  $  800,  how  much  does  he  owe  B  ? 

22.  1227.83  is  |-%  less  than  what  number? 

23.  4  is  20%  less  than  what  number? 


220  PERCENTAGE. 

24.  A  city  lot  cost  $3600,  which  is  55%  less  than  the 
cost  of  the  house.     What  was  the  cost  of  the  house  ? 

25.  A  farmer  raised  1500  bu.  of  corn,  which  was  33^% 
less  than  the  number  of  bushels  of  wheat  raised.  How 
many  bushels  of  wheat  had  he  ? 

26.  In  the  year  1896  a  merchant's  profits  were  $  1836.25, 
w^ich  was  25%  more  than  his  profits  of  1895.  What  were 
his  profits  in  1895  ? 


PROFIT  AND  LOSS, 

301.  Oral. 

State  the  question  only. 

1.  How  much  is  a  10%  profit  on  goods  that  cost  f  200? 

2.  I  bought  goods  for  $400,  and  sold  them  at  a  loss  of 
5%.     How  much  did  I  lose ? 

3.  If  I  buy  goods  at  f  400,  and  sell  them  at  $  600,  what 
per  cent  profit  do  I  make  ? 

4.  If  I  buy  at  $400,  and  sell  at  $350,  what  %  do  I 
lose  ? 

5.  By  selling  a  house  for  $1600  I  gain  33^-%.     What 

is  the  cost?  ? 

6.  John  sold  his  skates  for  64  cents,  and  thereby  lost     • 
5%.     What  did  he  pay  for  them  ? 

302.  All  computations   in  Profit  and  Loss  come  under 
the  rules  of  Percentage. 

The  cost  corresponds  to  the  base,  and  the  gain  or  4oss 
is  a  percentage  of  the  cost. 

The  selling  price  is  the  amount  when  there  is  a  profit, 
and  the  difference  when  there  is  a  loss. 


PROFIT   AND  LOSS.  221 

303.   Written. 

7.  If  I  buy  eggs  for  10  cents  a  doz.,  and  sell  them  for 
12^  cents,  what  per  cent  do  I  gain  ? 

8.  A  grocer  bought  tea  at  18  cents  per  pound,  and  sold 
it  at  30  cents  per  pound.     What  was  the  rate  of  gain  ? 

9.  Find  the  profit  on  a  bicycle  that  cost  f  75,  and  was 
sold  at  an  advance  of  30%. 

10.  Find  the  selling  price  of  a  horse  bought  at  $  88.65, 
and  sold  at  3J%  below  cost. 

11.  Find  the  rate  per  cent  of  loss  on  a  cow  bought  for 
$80,  and  sold  for  $60. 

12.  Find  the  rate  per  cent  of  profit  on  a  car-load  of 
Cortland  wagons  sold  for  $1090,  and  bought  for  $1000. 

13.  Find  the  cost  of  a  herd  of  cattle  sold  at  12|-%  above 
cost  at  a  profit  of  $  240. 

14.  A  man  bought  books  for  $194,  and  sold  them  at  a 
gain  of  32%.     What  was  the  gain  ? 

15.  I  sold  a  house  and  lot  that  cost  $11,225  at  a  loss  of 
5^%.     What  was  the  loss  ? 

16.  Mr.  A.,  by  selling  his  horse  at  a  profit  of  14%,  made 
$32.20.     What  did  the  horse  cost? 

17.  By  selling  sljgar  at  one-half  cent  per  pound  profit,  a 
grocer  makes Ife  per  cent.  What  does  he  get  per  pound 
for  his  sugar  ? 

18.  An  agent  gained  $.09  by  selling  twine  25%  above 
cost.     What  did  it  cost  him  ? 

19.  Find  the  cost  of  cotton  sold  at  16 1%  above  cost  at 
aprofit  of  $211.25. 


222  PERCENTAGE. 

20.  By  selling  flour  at  a  loss   of  14|^%,  a  grocer  loses 
$  13.45.     What  was  the  cost  ? 

21.  A  farm  that  cost  $2675  was  sold  for  $3745.     What 
was  the  gain  per  cent  ? 

22.  Hats  that  cost  $43.50  a  dozen  are  sold  for  $4.50 
apiece.     What  is  the  rate  of  gain? 

23.  By   selling   boots   for    $206.40   a   merchant   gained 
^.     What  did  they  cost  him  ? 


24.  By  selling  corn  for  $92.61,  a  man  gained  12^%. 
What  did  it  cost  him  ? 

25.  I  sell  a  horse  for  twenty  per  cent  less  than  my  ask- 
ing price,  and  yet  make  twenty-live  per  cent  profit.  I 
asked  $  200.     What  did  the  horse  cost  me  ? 

26.  My  height  is  6  feet  1^  inches,  my  neighbor  is  5  feet 
10  inches.     What  per  cent  am  I  taller  than  he  is  ? 

27.  A  farmer  sold  160  acres  of  land  for  $2944,  which 
was  8%  less  than  it  cost.     What  did  it  cost  an  acre  ? 

28.  By  selling  a  horse  for  $160,  I  lose  20%.  What 
would  have  been  the  selling  price  had  I  gained  20%  ? 

29.  14f  %  was  gained  by  selling  tea  at  $  .45  a  pound. 
What  did  it  cost  a  pound  ? 

30.  Mr.  Brown  sold  a  lot  for  $4300,  and  by  so  doing 
made  11^%.     What  did  he  gain  ? 

31.  If  I  buy  oranges  at  the  rate  of  3  for  3  cents,  and 
sell  them  at  the  rate  of  2  for  5  cents,  what  per  cent  profit 
do  I  make  ? 

32.  A  jeweller  sold  two  watches  at  $24  each.  On  one 
he  gained  20%,  and  on  the  other  he  lost  20%.  What  did 
both  watches  cost  him  ? 


COMMISSION.  223 

COMMISSION. 

304.  Oral. 

1.  A  certain  agent  receives  for  his  services  2%  of  the 
value  of  the  goods  which  he  sells.  How  much  will  he 
receive  for  selling  f  1000  worth  of  goods  ? 

2.  A  purchasing  agent  receives  for  his  services  3%  of 
the  value  of  the  goods  purchased.  How  much  will  he  receive 
for  purchasing  f  2000  worth  of  goods  ? 

.    3.    How  much  must  I  pay  my  agent  for  selling  f  3000 
worth  of  potatoes  if  I  pay  him  5%  ? 

4.  At  5%  how  much  will  a  collecting  agent  receive 
for  collecting  $800? 

5.  How  much  must  I  pay  my  broker  for  selling  $1000 
worth  of  stocks,  if  I  pay  him  1%  of  their  value  ? 

305.  An  Agent  is  a  person  who  transacts  business  for 
another. 

306.  Some  agents  are  known  as  Brokers,  or  Commission 
Merchants,  according  to  the  kind  of  business  transacted. 

307.  The  compensation  of  an  agent  is  called  Commis- 
sion, or  Brokerage. 

The  commission  of  a  purchasing  agent  is  usually  a  cer- 
tain per  cent  of  the  value  of  his  purchases. 

The  commission  of  a  sales  agent,  or  of  a  collector,  is 
usually  a  certain  per  cent  of  the  amount  collected. 

308.  The  merchandise  sent  to  a  commission  merchant  to 
be  sold  is  called  a  Consignment. 

309.  The  sender  is  the  Consignor,  and  the  person  to 
whom  the  goods  are  sent  is  the  Consignee. 

310.  The  commission  is  the  percentage,  and  the  amount 
collected  or  invested  is  the  base. 


224  PERCENTAGE. 

311.   Written. 

6.  An  auctioneer  charges  5%  commission  for  selling 
$864  worth  of  goods.  What  is  the  amount  of  his  com- 
mission ? 

7.  Sold  850  barrels  of  flour  at  $5.25  a  barrel,  and 
charged  2^%  commission.     Find  my  commission. 

8.  What  is  an  agent's  commission  for  selling  6840  lb.  of 
butter,  at  19  cents  a  pound,  commission  l|-%  ? 

9.  A  dealer  sells  real  estate  for  a  commission  of  2%. 
How  much  must  he  sell  during  the  year  to  secure  an 
income  of  $75  per   month? 

10.  A  broker  in  New  York  received  J  of  one  per  cent 
commission  for  negotiating  a  sale  of  500  one  thousand 
dollar  bonds.     What  was  his  commission? 

11.  A  real  estate  man  made  $50  by  receiving  2 J  per 
cent  instead  of  his  regular  commission  of  2  per  cent. 
What  did  his  sales  amount  to? 

12.  An  agent's  fee  for  collecting  bills  is  3%.  If  he 
receives  $86.25  as  his  commission,  how  much  money  has 
he  collected  ? 

13.  An  agent  collected  $1864  from  a  sale  of  some 
pictures,  and  received  $  4.66  as  his  fee.  What  was  the 
rate  of  commission  ? 

14.  An  agent  having  sold  1250  velocipedes  at  $8  apiece, 
invested  his  commission  of  If  %  in  a  new  stock  company. 
How  many  shares  at  $  25  each  did  he  take  ? 

15.  What 'per  cent  does  an  agent  charge  who  receives 
$223  for  buying  $5575  worth  of  produce  ? 

16.  A  man  sends  his  agent  $6120  to  invest  in  flour, 
after  deducting  his  commission  of  2%.  How  much  money 
is  spent  for  the  flour,  and  how  much  for  the  agent's 
commission  ? 


COMMISSION.  225 

17.  A  man  is  paid  5%  for  collecting  $235.75,  How 
much  must  he  pay  over  to  his  employer? 

18.  A  merchant  sent  his  agent  $3150  with  which  to  buy 
flour  after  deducting  his  commission  of  5%.  At  $4  per 
barrel  how  many  barrels  did  the  agent  buy  ? 

19.  An  agent  sold  iron  for  $9872.  He  received  $  163.70, 
which  included  a  freight  charge  of  $52.64.  What  rate  of 
commission  did  he  receive? 

20.  Received  as  net  proceeds  from  a  sale  of  cotton 
$1025.70,  after  paying  my  agent  2i%  for  selling.  What 
did  the  sale  amount  to  ? 

21.  An  auctioneer  sells  15  tables  at  $1.45  apiece,  22 
chairs  at  $  1.12^  ajjiece,  and  some  pictures  for  $  8.70,  on 
a  commission  of  5|-%.  What  were  his  commission  and  the 
net  proceeds  of  the  sale  ? 

22.  My  agent  in  Boston  sold  a  number  of  bicycles  at 
$85  each.  After  deducting  his  commission  of  3|-%,  he 
returned  to  me  $  5759.60.     How  many  bicycles  did  he  sell  ? 

23.  An  agent  who  sold  150  lots  at  $  233J  each,  charged 
$  262.50  for  his  services.  What  rate  of  commission  did  he 
get? 

24.  A  collector  pays  over  to  his  principal  $23358.39^, 
after  deducting  a  commission  of  4-|^%.  How  much  was  the 
entire  collection  ? 

25.  If  I  send  my  agent  $  367.20,  with  instructions  to  buy 
tea  at  30  ct.  a  pound,  and  he  charges  2%  for  buying,  how 
many  pounds  of  tea  should  I  receive  ? 

26.  A  real  estate  agent  charges  me  two  per  cent  for  sell- 
ing my  property  in  Boston.  He  remits  me  $  5880.  What 
was  his  commission  ? 


226  PERCENTAGE. 

27.  A  commission  merchant  in  Kew  York  charged  $36 
foi:  insuring  my  goods,  $  14  for  cartage,  and  $  50  commis- 
sion at  2i  per  cent  for  selling  them.  How  much  money 
should  he  remit  to  me  ? 

28.  Sent  my  agent  $  2050  to  invest  in  coal  at  $  4  per 
ton,  after  deducting  his  commission  of  2i  per  cent.  How 
many  tons  of  coal  could  he  buy  ? 

29.  A  cotton  broker  received  $  2531.71  with  which  to  buy 
cotton  at  $  .12  a  pound.  He  charged  2^  %  commission. 
How  many  pounds  of  cotton  did  he  buy,  and  what  was  his 
commission  ? 

30.  A  real  estate  agent  receives  $  162,193.50  from  a  com- 
pany to  invest  in  land.  If  he  charges  5%  commission,  how 
many  acres  of  land  can  he  buy  at  $9  an  acre  ?  What  is 
his  commission  ? 

31.  An  agent  sold  12000  lb.  of  cotton  at  10  cents  a 
pound.  He  invested  the  proceeds  in  lumber  at  $  25  per  M. 
If  his  commission  for  selling  was  4%,  and  for  buying  2%, 
how  many  feet  of  lumber  did  he  purchase  ? 

32.  A  grain-dealer  received  $4820.40  with  which  to  buy 
wheat  at  60^  a  bushel  after  deducting  his  commission  of 
3%.     How  much  wheat  did  he  purchase  ? 

33.  How  much  stock  can  be  bought  for  $  10827,  allowing 
li%  brokerage  ? 

34.  A  speculator  sent  $  7308  to  his  agent  in  St.  Paul, 
directing  him  to  invest  in  wheat.  How  many  bushels  of 
wheat  could  he  buy  at  90^  a  bushel,  after  deducting  his 
commission  of  1^%  ? 

35.  An  agent  has  bought  1170  bbl.  of  flour  at  $  4.50  per 
barrel.  How  much  money  must  his  principal  remit  to  him 
to  pay  the  cost  of  the  flour  and  his  commission  of  2%? 


INSURANCE.  227 


INSURANCE. 


312.  Insurance  is  security  against  loss.  The  insurance 
company  agree,  for  a  specified  amount,  to  be  paid  at  stated 
periods  (usually  once  a  year),  to  pay  a  definite  sum  to  the 
insured  or  his  estate  in  case  of  loss.  Some  of  the  different 
forms  of  insurance  are  Life,  Fire,  and  Accident. 

313.  The  stated  sum  paid  for  insurance  is  called  the 
Premium.     It  is  always  a  certain  per  cent  of  the  insurance. 

314.  The  written  contract  between  the  insurance  company 
and  the  insured  is  called  the  Policy. 

315.  Life  Insurance.  The  following  are  the  principal 
forms  of  policies  issued  by  life  insurance  companies : 

316.  Life  Policies,  with  continuous  premiums,  or  with 
premiums  limited  to  5,  10,  15,  or  20  years ;  also,  single  pay- 
ment life.     These  policies  are  payable  at  death  only. 

317.  Endowment  Policies,  with  continuous  premiums  for 
10,  15,  20,  25,  30,  35,  or  40  years;  also,  single  payment 
endowment  for  either  of  the  periods  mentioned,  and  10 
payment  endowment  for  15,  20,  25,  or  30  years.  These 
policies  are  payable  at  the  end  of  the  stated  period,  or  at 
death,  if  it  occurs  before  the  end  of  the  period. 

There  are  various  other  forms  of  policies,  but  the  above 
are  the  most  common. 

1.  For  an  annual  premium  of  $22.63  a  certain  company 
will  insure  a  man  30  years  of  age  for  $1000,  payable  at* 
death  (Life  Policy).  The  premium  might  be  stated  as 
2.263%.  What  would  a  $5000  life  policy  cost  the  man 
each  year  ? 


228 


PERCENTAGE. 


The  following  table  may  be  used  to  determine  the  premium 
rates  per  $  1000  for  ages  20-44  inclusive : 


Whole  Life  Policies. 

Endowments. 

i 

< 

20 

IS 

■^   o 
^1  fe 

S 

m 

hi 

hi 

O 
< 

20 

24  33 

28  83 

3811 

296  05 

100  00 

6314 

45  29 

35  05 

28  62 

21 

17  70 

24  82 

29  39 

38  84 

30154 

10011 

63  27 

45  43 

35  21 

28  79 

21 

22 

18  15. 

25  32 

29  97 

39  60 

307  21 

100  23 

63  40 

45  58 

35  37 

28  97 

22 

23 

18  62 

25  84 

30  58 

40  39 

313  08 

100  35 

63  54 

45  74 

35  54 

29  17 

23 

24 

1911 

26  38 

3121 

4121 

319  14 

100  49 

63  69 

45  90 

35  72 

29  39 

24 

25 

19  63 

26  95 

3187 

42  05 

325  41 

100  63 

63  84 

46  07 

35  91 

29  63 

25 

26 

20  17 

27  54 

32  55 

42  93 

33189 

100  78 

64  01 

46  25 

36  12 

29  88 

26 

27 

20  74 

28  15 

33  25 

43  84 

338  58 

100  93 

6418 

46  44 

36  35 

3014 

27 

28 

2134 

28  78 

33  i)8 

44  78 

345  50 

10110 

64  37 

46  65 

36  59 

30  42 

28 

29 

2197 

29  44 

34  74 

45  75 

352  64 

10127 

64  56 

46  87 

36  85 

30  73 

29 

30 

22  63 

3012 

35  53 

46  76 

360  02 

10145 

64  76 

4710 

3713 

3107 

30 

31 

23  32 

30  83 

36  34 

47  81 

367  04 

10164 

64  98 

47  35 

37  43 

3144 

31 

32 

24  05 

3158 

37  19 

48  89 

375  51 

10184 

65  20 

47  62 

37  76 

3183 

32 

33 

24  82 

32  36 

38  07 

50  01 

383  63 

102  00 

65  44 

47  92 

3812 

32  25 

33 

34 

25  63 

3317 

38  99 

5117 

392  02 

102  28 

65  71 

48  24 

38  50 

32  71 

34 

35 

26  49 

34  01 

39  94 

52  38 

400  68 

102  51 

65  99 

48  58 

38  92 

33  21 

35 

36 

27  39 

34  90 

40  93 

53  64 

409  63 

102  76 

66  29 

48  95 

39  37 

33  76 

36 

37 

28  35 

35  83 

4197 

54  94 

418  87 

103  03 

66  62 

49  36 

39  87 

34  36 

37 

38 

29  36 

36  81 

43  06 

56  29 

428  42 

103  33 

66  99 

49  82 

40  42 

35  01 

38 

39 

30  43 

37  84 

44  20 

57  70 

438  29 

103  65 

67  40 

50  32 

4102 

35  73 

39 

40 

3157 

38  92 

45  39 

5917 

448  49 

104  01 

67  85 

50  87 

4168 

36  52 

40 

41 

32  78 

40  07 

46  65 

60  71 

459  05 

104  41 

68  34 

5148 

42  41 

37  38 

41 

42 

34  07 

4129 

47  97 

62  33 

469  97 

104  86 

68  89 

52  15 

43  22 

38  32 

42 

43 

35  45 

42  58 

49  37 

64  02 

48126 

105  36 

69  51 

52  88 

4410 

39  35 

43 

44 

36  91 

43  94 

50  84 

65  79 

492  88 

105  92 

70  20 

53  71 

45  07 

40  48 

44 

2.  For  an  annual  premium  of  ^47.10  the  same  company- 
will  insure  the  same  man  for  $  1000,  payable  in  20  years,  or 
at  death,  if  the  insured  dies  before  20  years  (20-year  endow- 
ment policy).  What  would  a  $2500  20-year  endowment 
policy  cost  the  man  each  year  ? 


3.  What  is  the  premium  on  a  life  policy  for  f  2500,  age 
37,  continuous  annual  premium  ?  On  a  10-payment  life 
policy  ?  On  a  15-year  endowment  ?  On  a  single-payment 
life  policy  ? 


INSURANCE. 


229 


4.  A  man  25  years  of  age  is  insured  for  $  5000,  20-pay- 
ment  life  policy.  He  dies  at  the  age  of  34.  How  much 
more  is  paid  to  his  estate  than  his  insurance  cost  him  ? 

5.  What  will  be  the  cost  of  a  10-year  endowment  policy 
of  $3000,  age  20,  on  the  annual  dividend  plan,  the  first 
dividend  of  $  10.50  per  $  1000  being  paid  when  the  second 
premijim  is  due,  each  succeeding  dividend  increasing  1% 
of  the  premium,  the  last  one  being  paid  at  the  end  of  the 
11th  year  ? 

soiiimoN. 


Age  20. 

Dividend. 

Net  Cost. 

4 

1 

10 
11 

31.50 
34.50 
37.50 
40.50 
43.50 
46.50 
49.50 
52.50 
55.50 
58.50 

300.00 
268.50 
265.50 
262.50 
259.50 
256.50 
253.50 
250.50 
247.50 
244.50 

2608.50 

—  58.50 = 2550.00  =  cost. 


If  the  full  premium  of  $  300  is  paid  each  of  the  ten  years, 
and  the  profits  are  allowed  to  accumulate  so  that  at  maturity 
(i.e.  at  the  end  of  the  ten  years)  the  policy  is  worth  $  3627 
1$  3000  being  the  face  value  of  the  policy,  and  $  627  the 
accumulated  profits),  what  per  cent  is  paid  on  the  money 
invested  ?  This  form  of  insurance  is  sometimes  called 
Semi-Tontine  Insurance. 


318.  Annual  Dividend.  Since  it  is  impossible  to  exactly 
foretell  the  actual  cost  of  insurance,  it  is  necessary  to  make 
the  premiums  large  enough  to  meet  all  contingencies,  such 
as  increased  death  rate  and  reduced  rate  of  interest.  Con- 
sequently, there  usually  arises  a  surplus,  caused  by  over- 


230  PERCENTAGE. 

payments.  This  surplus  is  distributed  annually  to  each 
policy  holder  in  proportion  to  his  over-payment,  and  con- 
stitutes the  Annual  Dividend. 

319.  Semi-Tontine.  Under  this  plan  the  surplus,  instead 
of  being  paid  annually  as  dividends,  is  allowed  to  remain 
with  the  company  at  interest,  and  accumulate  for  a  period 
of  10,  15,  or  20  years,  as  may  be  selected.  Those  who  sur- 
vive, and  keep  their  policies  in  force  until  the  end  of  the 
period,  receive  both  the  surplus  arising  from  their  own 
policies,  and  their  equitable  share  of  the  surplus  arising 
from  policies  discontinued  by  lapse  or  death  during  the 
period. 

320.  Fire  Insurance  is  security  against  loss  by  fire. 

321.  1.  I  keep  my  house  insured  for  $4000.  I  pay  the 
insurance  company  1%  annually.  How  much  do  I  pay 
annually  ? 

2.  If  my  house  (Ex.  1)  burns  down  at  the  end  of  three 
years,  how  much  shall  I  receive  from  the  company  more 
than  I  have  paid  them  ? 

3.  If  I  pay  $75  per  annum  for  insuring  my  house  at 
1%,  for  how  much  is  it  insured? 

4.  At  2%  what  will  be  the  cost  of  insuring  $10000  of 
merchandise  at  |  value  ? 

5.  My  household  goods  are  insured  at  the  rate  of  $.45 
per  annum  on  each  $100.  What  premium  do  I  pay  if  the 
amount  of  insurance  is  $  1200  ? 

6.  For  a  premium  of  $  .90  on  each  $  100  the  same  com- 
pany will  insure  household  goods  for  a  period  of  three 
years.  How  much  is  saved  on  $1200  insurance  on  this 
plan,  as  compared  with  the  plan  in  Example  6  ? 


INSURANCE.  231 

322.  Accident  Insurance.  The  following  is  one  form  of 
accident  insurance :  For  a  quarterly  premium  of  $  5,  a 
person  may  be  insured  for  $  10000  in  case  of  death,  or  a 
weekly  indemnity  not  to  exceed  $50  will  be  paid,  not  to 
exceed  52  weeks,  in  case  of  total  disability  caused  by  acci- 
dent while  a  passenger  on  a  public  conveyance  propelled  by 
steam,  electricity,  or  cable.  One  half  the  above  amounts 
will  be  paid  in  case  of  an  accident  due  to  any  other  cause. 

323.  1.  After  paying  seven  such  premiums  a  person  is 
thrown  from  his  bicycle,  and  injured  so  that  he  is  disabled 
for  5  weeks.  How  much  more  does  he  receive  than  he  has 
paid  out  ? 

2.  If  he  had  been  injured  in  a  railroad  accident,  and  dis- 
abled for  the  same  time,  how  much  more  would  he  have 
received  ? 

3.  What  will  be  the  premium  on  a  30-year  endowment 
policy,  age  29,  for  $4000  ? 

4.  If  the  total  dividends  amount  to  30%  of  the  total 
premiums  paid,  how  much  more  does  the  man  receive  at 
the  end  of  the  30  years  than  he  has  paid  ? 

5.  What  will  it  cost  to  insure  a  house  worth  $2500  at 
|-  of  its  value,  for  3  years,  at  f  %  ? 

6.  Injured  a  country  store  for  $  5000,  and  goods  for 
$10,000,  at  30^  on  $100.  $1  is  paid -for  the  policy. 
What  does  the  insurance  cost  ? 

7.  What  will  it  cost  to  insure  a  mill  for  $5000,  the  rate 
being  one  and  one-half  per  cent  for  3  years  ? 

8.  How  much  will  I  save  by  insuring  my  property  for 
$  5000  at  f  of  one  per  cent  for  3  years,  rather  than  taking 
an  annual  policy  for  i  of  one  per  cent  ? 


232  PERCENTAGE. 

9.    Tt  costs  me  to  insure  my  house  f  22.50  when  the  rate 
is  J  of  one  per  cent.     What  is  the  amount  of  my  policy  ? 

10.  A  stock  of  goods  is  insured  for  one-half  the  value, 
the  premium  being  $  30,  and  the  rate  y^  of  one  per  cent. 
What  is  the  value  of  the  goods  ? 

11.  The  semi-annual  premium  per  one  thousand  dollars 
on  my  $6000  life-insurance  policy  is  $26.  What  does  it 
cost  me  a  year  ? 

12.  A  person  who  pays  $  12  semi-annually  for  accident 
insurance  is  disabled  by  an  accident  for  13  weeks,  during 
which  time  he  receives  $  10  a  week.  If  he  had  paid  three 
premiums,  how  much  more  does  he  receive  than  he  has 
paid  out  ? 

13.  Paid  for  insuring  a  house  for  ^  of  its  value,  $151. 
The  rate  being  75^  on  $100,  and  the  policy  costing  $1, 
what  was  the  house  worth  ? 

14.  To  insure  a  house  at  -J  of  1%  cost  me  $20.  What 
was  the  house  worth  ? 

15.  Paid  $18  for  insuring  goods  worth  $9000.  What 
was  the  rate  ? 

16.  A  merchant  pays  $75  a  year  insurance  on  his  stock 
of  goods  at  1J%.     What  is  the  value  of  his  stock  of  goods? 

17.  A  block  worth  $30000  is  insured  for  |  of  its  value 
at  2%.  How  much  does  the  owner  lose  in  case  of  its  total 
destruction  by  fire  ? 

18.  For  how  much  must  a  cargo  of  wheat  worth  $23400 
be  insured,  at  2i%,  so  that  the  owner,  in  case  of  loss,  may 
recover  both  the  value  of  the  cargo  and  the  premium  ? 

Note.  — The  value  of  tlie  wheat  =  97^%  of  the  amount  insured. 


TKADE  DISCOUNT.  233 


TRADE  DISCOUNT. 


^<^i.2ms^^ 


324.  The  deduction  of  a  percentage  from  the  price  of 
merchandise  is  called  Commercial  Discount. 

It  is  used  largely  by  manufacturers  and  wholesale  mer- 
chants. The  greatest  discounts  are  for  large  purchases 
and  cash  payment. 

325.  The  List  Price  is  the  price  given  in  the  price-list. 

326.  The  Net  Price  is  the  list  price  less  the  discount. 

1.  If  I  can  purchase  books  at  25%  off  for  cash,  what 
must  I  pay  for  books  listed  at  ^  80  ? 

Solution.  — 100  %  -  25  %  =  75  %.     75  %  of  $  80  =  $  60.    Ans. 

2.  At  what  per  cent  above  cost  must  a  merchant  mark 
his  goods  so  that  he  may  allow  a  discount  of  25%  from  the 
marked  price,  and  still  make  a  profit  of  10%  ? 

Solution.  —  Selling  price  =  110%  of  cost.  This  selling  price  is  75% 
of  the  marked  price.  The  question  is,  "110%  is  75%  of  what?" 
1.10  -=-  .75  =  1.46|  %,  therefore  the  marked  price  is  46f  %.above  cost. 

3.  Find  the  sum  to  be  paid  on  a  bill  of  $264  with  10% 
off  for  cash. 

4.  What  is  the  net  price  of  a  bill  of  goods,  the  list  price 
of  which  is  $56,  subject  to  discount  of  25%  ? 

5.  What  must  be  paid  on  $935,  if  15%  and  10%  off  are 
allowed  ? 

Solution. — Deducting  15%  is  the  same  as  allowing  85%  of  the 
bill.     85  %  of  935  =  794.75.     90  %  of  794.75  =  715.28.     Ans. 

Note.  —  When  two  or  more  discounts  are  allowed,  the  first  is 
deducted,  the  second  computed  on  the  remainder,  and  deducted  from 
it,  etc. 

6.  Which  is  the  better  for  the  buyer,  40%,  or  25%  and 
15%  off? 


234  PERCENTAGE. 

7.  Find  a  single  discount  on  a  bill  of  $300  equal  to 
20%  and  5%  ofp. 

8.  A  discount  of  $4  was  allowed  on  a  bill,  which  was 
then  paid  with  a  check  for  $36.-  What  rate  per  cent  was 
taken  off? 

9.  Consulting  my  price-list,  I  find  I  can  buy  goods 
which  are  marked  $450  at  a  discount  of  20%  and  5%  olf 
for  cash.  How  much  will  the  goods  cost  me,  and  how 
much  discount  do  I  receive  ? 

10.  Bought  furniture  amounting  to  $520  on  credit  for, 
6  months,  or  5%  "discount  for  cash.  What  ready  money 
will  pay  the  bill? 

11.  What  is  the  cash  value  of  a  bill  of  books  amounting 
to  $40,  on  the  face  of  which  a  discount  of  20%  and  5%  is 
made  ? 

12.  The  net  amount  of  a  bill  of  goods  is  $  359.10.  What  is 
the  gross  amount,  the  rate  of  discount  being  10%  and  5%  ? 

13.  A  set  of  Encyclopaedias,  whose  catalogue  price  is 
$100,  can  be  bought  at  a  discount  of  2  tens  and  5%  off  for 
cash.  How  much  less  than  the  catalogue  price  will  they 
cost? 

Note.  —  The  expression  2  tens  and  5%  means  10%,  10%,  and  5%. 

14.  B  offers  me  some  hammocks  for  $450  with  a  dis- 
count of  20%,  and  4%  off  for  cash,  and  A  offers  me  the 
same  goods  at  a  discount  of  2  tens  and  4%  off.  Which  is 
the  better  offer,  and  how  much  ? 

15.  A  dealer  sold  goods  at  10%  below  his  asking  price, 
but  still  made  a  profit  of  20%.  What  per  cent  above  cost 
had  he  marked  the  goods  ? 

16.  A  merchant  marked  carpeting  that  cost  him  60  cents 
a  yard  so  that  he  could  allow  a  discount  of  10%  and  still 
make  a  profit  of  20%.     At  what  price  did  he  mark  it? 


TAXES.  2B5 

17.  A  book-dealer  sold  a  stock  of  books  for  $  1140,  at  a 
discount  of  10%  from  the  marked  price,  and  finds  that  he 
has  made  a  profit  of  14%.  What  did  he  pay  for  the  books, 
and  what  was  their  marked  value  ? 

18.  Find  the  net  amount  of  a  bill  for  $386  subject  to 
the  following  discounts,  20%,  10%,  and  5%. 

TAXES. 

327.  A  tax  is  a  sum  of  money  levied  upon  property  and 
persons  for  public  use. 

Note. — A  tax  upon  persons  is  called  Capitation  or  Poll  Tax. 
It  is  levied  in  some  localities  upon  men  of  full  age,  without  regard 
to  their  property.  It  is  usually  but  a  small  amount  upon  each 
person.     The  practice  is  going  out  of  use. 

328.  Property  is  of  two  kinds,  Real  and  Personal. 

329.  Eeal  Property  is  immovable  property,  as  lands 
and  buildings. 

330.  Personal  Property  is  property  that  is  movable,  as 
money,  securities,  household  goods,  horses,  cattle,  etc. 

331.  A  tax  assessed  upon  property  is  a  Property  Tax. 

332.  Assessors  are  officers  chosen  to  make  a  list  of  tax- 
able property,  estimate  its  value,  and  apportion  the  tax. 

333.  A  tax  is  a  percentage  upon  the  assessed  valuation 
of  property.     The  tax  on  $  1  is  the  rate. 

1.  The  valuation  of  property  in  a  certain  town  is 
$1,500,000,  and  the  rate  is  11%?.     What  is  the  tax? 

2.  The  tax  to  be  raised  in  a  certain  village  is  $37,500. 
The  taxable  property  is  $2,500,000.  What  is  the  rate? 
What  will  be  A's  tax  on  $15,000  real  estate,  and  $3000 
personal  ? 


236  PERCENTAGE 

3.  The  property  of  a  town  is  assessed  at  $1,250,000. 
The  tax  to  be  raised  is  $15,975.  There  are  650  polls, 
assessed  at  $1.50  each.  What  is  B's  entire  tax,  if  his 
property  is  assessed  at  $2500,  and  he  pays  the  poll- 
tax? 

Rule.  —  Deduct  the  amount  of  poll-tax,  if  any,  from  the 
whole  tax.  Divide  the  remainder  by  the  assessed  valu- 
ation. The  quotient  will  be  the  rate. 
To  find  each  person's  tax,  midtiply  the  assessed  valua- 
tion by  the  rate,  and  to  the  product  add  the  poll-tax^ 
if  any. 

4.  The  officers  of  a  certain  town  find  that  all  the  town 
expenses  for  the  year  1896  will  amount  to  $46,000.  The 
tax-roll  shows  real  estate  valued  at  $2,000,000,  and  per- 
sonal property  at  $300,000.  What  is  the  rate  of  taxa- 
tion? 

5.  A  certain  town  votes  to  raise  a  tax  of  $14,250,  be- 
sides the  collector's  commission  of  5%.  What  is  the  rate 
of  taxation  if  the  property  valuation  is  $  1,000,000  ? 

What  is  the  collector's  commission,  and  what  is  A's  tax, 
on  property  valued  at  $  4500  ? 

6.  If  the  assessed  valuation  of  a  village  is  $2,384,564, 
and  there  are  750  polls  at  $  1.50  each,  what  must  be  the 
rate  of  taxation  to  meet  an  expense  of  $29,807.05?  What 
is  B's  entire  tax,  if  his  property  is  valued  at  $3875,  and  he 
pays  for  1  poll  ? 

7.  What  is  the  valuation  of  my  property,  if  my  tax,  15 
mills  on  a  dollar,  amounts  to  $  30  ? 

8.  What  is  my  entire  tax,  if  I  pay  a  poll-tax  of  $1.68, 
and  my  property  is  valued  at  $24,750,  when  the  rate  of 
taxation  is  $16.28  on  $1000? 


DUTIES.  237 

9.  The  annual  tax-rate  for  the  State  of  New  York  for 
the  year  1896  was  2.69  mills  on  the  dollar.  The  amounts 
to  be  raised  by  tax  are  as  follows :  $  961,116  for  general 
expenses,  §4,062,903  for  free  schools,  §2,360,103  for  the 
canals,  and  §4,368,712  for  the  State  care  of  the  insane. 
What  was  the  assessed  valuation  of  the  property  of  the 
entire  State? 

The  tax-rate  for  1895  was  3.24  mills.  What  was  the 
entire  tax  of  1895? 

DUTIES. 

334.  Duties  are  taxes  on  imported  goods,  levied  by  the 
government,  and  collected  at  custom-houses.  A  port  con- 
taining a  custom-house  is  called  a  Port  of  Entry. 

335.  An  Ad  Valorem  Duty  is  a  certain  rate  on  the  value 
of  goods  at  the  place  from  which  they  were  shipped. 

336.  A  Specific  Duty  is  a  fixed  sum  charged  upon  an  im- 
ported article,  without  regard  to  its  value.  Allowances 
are  made  as  follows,  in  collecting  specific  duties :  for  Tare, 
which  is  weight  of  box,  cask,  etc. ;  for  Leakage,  which  is 
loss  of  liquids  in  barrels  or  casks ;  and  for  Breakage,  which 
is  loss  of  liquids  in  bottles. 

337.  The  Gross  Weight  is  the  weight  of  articles  before 
any  allowances  are  made. 

338.  The  Net  Weight  is  the  weight  after  the  allowances 
are  made. 

1.  At  30  per  cent  ad  valorem,  what  is  the  duty  on  goods 
valued  at  §725? 

2.  What  is  the  duty  on  10  gross  of  silver  spoons,  valued 
at  §4.50  a  dozen,  at  30%  ad  valorem? 


238  PERCENTAGE. 

3.  A.  Mark's  Sons  imported  from  Lyons  1560  yd.  of  silk 
invoiced  at  87^  ^  per  yard.  What  was  the  duty  at  25  ^  a 
yard,  and  30%  ad  valorem  ? 

4.  If  the  average  rate  of  duty  under  the  McKinley  law 
was  49.58  per  cent,  and  under  the  Wilson  law  it  is  37  per 
cent,  what  is  the  difference  in  revenue  on  $  1,000^000  worth 
of  dutiable  imports  ? 

5.  A  merchant  bought  goods  in  London  invoiced  at 
£>  450.  At  the  custom-house  in  New  York  he  paid  an  ad 
valorem  duty  of  18%,  and  a  specific  duty  of  $325.  What 
was  the  entire  cost  of  the  goods  in  United  States  money  ? 

6.  Imported  from  England  5  cases  of  cloths  and  cash- 
meres, net  weight  95  lb. ;  value  as  per  invoice  £  375  10s. 
What  is  the  duty,  the  rate  being  50^  per  pound,  and  35% 
ad  valorem  ? 

QUESTIONS. 

339.  1.  What  is  the  meaning  of  the  term  per  cent? 
How  is  per  cent  written? 

2.  Define  Base,  Percentage,  Rate  per  cent.  Amount,  Dif- 
ference. 

3.  Tell  how  to  find  percentage  when  base  and  rate  are 
given.  To  find  base  when  percentage  and  rate  are  given. 
To  find  rate  when  percentage  and  base  are  given. 

4.  Tell  how  to  find  base  when  amount  and  rate  are 
given.     When  difference  and  rate  are  given. 

5.  Define  Commission,  Brokerage,  Insurance,  Premium, 
Policy,  Taxes,  Eeal  Estate,  Personal  Property. 

6.  What  is  trade  discount?  List  price?  Net  price? 
Give  the  rule  for  finding  net  price. 


MISCELLANEOUS   REVIEW.  239 

MISCELLANEOUS  REVIEW   OF  PERCENTAGE. 

340.  1.  Find  8%  of  750.  Q>\%  of  ^12.75.  |%  of  912. 
-{^%  of  2140.      • 

2.  A  man  gave  his  son  42%  of  his  money,  his  daughter 
25%  of  it,  and  his  wife  16-|%  of  the  remainder.  If  the 
son  received  $  9350  more  than  the  daughter,  what  did  each 
receive  ? 

3.  A  dealer  sold  a  horse  and  carriage  for  $  637,  which 
was  40%  more  than  cost.  If  the  horse  cost  -f-  as  much  as 
the  carriage,  what  did  each  cost  ? 

4.  What  per  cent  is  gained  or  lost  when  one-half  an 
article  is  sold  for  what  the  whole  cost?  When  f  of  an 
article  is  sold  for  what  one-half  cost  ? 

5.  A  merchant  pays  $  35  for  a  suit  of  clothes.  What 
must  he  ask  for  it,  so  that  he  may  drop  16%  from  his 
asking  price,  and  still  make  20%  on  the  cost? 

6.  14.35  is  j7^%  of  what  number? 

7.  A  man  spent  20%  of  his  salary  for  board  and  15% 
of  what  was  left  for  clothes.  If  he  spent  $  132  more  for 
board  than  for  clothes,  how  much  did  he  spend  for  each  ? 

Ni       8.    What  number  diminished  by'16|%  is  605  ? 

9.    What  number  increased  35%  is  382.5? 

10.  Sold  a  load  of  wheat  weighing  3240  lb.  at  68^  a 
bushel  of  60  lb.,  thereby  making  a  profit  of  Q>\  per  cent. 
Required  the  cost  of  the  wheat. 

11.  On  a  certain  day  the  sun  rose  at  5  o'clock  and  43 
minutes,  and  set  at  6  o'clock  and  25  minutes.  What  per 
cent  of  the  day  was  in  sunlight  ? 

12.  The  salary  of  a  certain  teacher  of  arithmetic  is 
$  1600.  His  real  estate  tax  is  $  90 ;  his  water  tax  is  $  25  ; 
gas  bill,  ^  15 ;  coal  bill,  ^  45  ;  other  expenses,  $  325.  What 
per  cent  of  his  salary  does  he  save  ? 


240  PERCENTAGE. 

13.  The  Oswego  Starch  Factory  employs  700  operatives. 
The  population  of  Oswego  numbers  22000.  What  per  cent 
of  the  population  are  employed  in  the  starch  factory  ? 

14.  I  is  25%  more  than  what  fraction? 

15.  A  dealer  lost  10%  of  his  capital,  then  gained  20% 
of  the  remainder,  when  he  had  $  2160.  How  much  had  he 
at  first? 

16.  Goods  bought  for  $400  are  marked  to  sell  at  an 
advance  of  40%,  but  are  finally  sold  at  a  reduction  of  25% 
from  the  marked  price.  What  is  the  per  cent  of  gain? 
What  is  the  gain  ? 

17.  An  article  is  sold  for  $  2.80,  this  being  an  advance  of 
2^0/0.     Find  the  cost. 

18.  A  merchant  buys  sugar  at  an  average  price  of  4 
cents  a  pound,  and  sells  at  a  profit  of  8%.  How  many 
pounds  must  he  sell  to  clear  %  500  ? 

19.  If  by  selling  an  article  for  59  cents  a  dealer  gains 
10%  more  than  by  selling  for  55  cents,  what  is  the  original 
cost? 

20.  15%  of  an  estate  is  invested  in  city  bonds,  40% 
in  real  estate,  25%  in  railroad  stock,  and  the  remainder, 
%  5000,  is  deposited  in  a  bank.     What  is  the  estate  worth  ? 

21.  Define  Percentage,  Base,  Profit  and  Loss,  Commission. 

22.  Give  the  five  formulas  of  percentage. 

23.  Express  as  a  decimal  f  per  cent. 

24.  A  man  having  a  yearly  income  of  %  1500  spends 
80%  of  it  the  first  year,  75%  of  it  the  second  year,  ^2\clo 
of  it  the  third.     How  much  does  he  save  in  3  years  ? 

25.  25%  of  200  bushels  is  2i%  of  how  many  bushels  ? 

26.  A  man  sold  80  acres  of  land  for  %  1472,  which  was 
8%  less  than  it  cost.     What  did  it  cost  an  acre  ? 


MISCELLANEOUS   KEVIEW.  241 

27.  What  terms  in  Profit  and  Loss  correspond  to  base 
and  amount  ? 

28.  Find  the  cost  of  fruit  sold  for  $  207.70,  at  a  gain  of 
15%? 

29.  At  what  price  must  hats  that  cost  $  1.12  each  be 
marked  in  order  to  abate  5%,  and  yet  make  25%  profit  ? 

30.  What  is  the  base  in  commission  ? 

31.  A  commission  merchant  sells  225  bu.  of  corn  at  $  .65 
a  bushel,  and  360  bbl.  of  apples  at  $  2.40  per  barrel,  com- 
mission 5%.     Find  the  commission  and  the  net  proceeds. 

32.  The  net  proceeds  are  $3800,  the  rate  10%.  Find 
the  amount  of  sales  and  the  commission. 

33.  Find  the  rate,  the  commission  being  $  125,  and  the 
sum  invested  $  2500. 

34.  A  merchant  owning  f  of  a  cargo  valuld  at  $  44000 
insures  f  of  his  share  at  2^%.  What  premium  does  he 
pay? 

35.  A  man  having  $  400  paid  62^%  of  it  for  a  carriage. 
How  many  dollars  had  he  left  ? 

36.  An  agent  charged  $  432.46  for  selling  goods  at 
$  49424.     What  was  his  rate  of  commission  ? 

37.  A  man  sold  four  horses  for  $  100  each.  On  two  he 
gained  25fcy  and  on  the  other  two  he  lost  25%.  Did  he 
gain  or  lose  on  the  transaction,  and  how  much  ? 

38.  If  f  the  number  of  girls  in  a  certain  school  exceed 
the  boys  10%,  and  the  girls  number  275,  what  is  the  num- 
ber of  boys  ? 

39.  A  farmer's  sheep  increased  10%  each  year  for  2 
years,  when  he  had  242.     How  many  had  he  at  first? 


242  PERCENTAGE. 

40.  My  New  York  agent  buys  for  me  40  pieces  of  silk, 
32  yards  in  a  piece,  at  $  5  a  yard.  He  charges  1^%  com- 
mission. How  much  money  will  it  require  to  purchase  the 
silk  and  pay  his  commission  ? 

41.  A  commission  merchant  in  Boston  has  sold  goods 
for  me  to  the  amount  of  $6932.  He  has  charged  l\-% 
commission,  $  18.50  cartage,  and  $  12.15  for  storage.  How 
much  is  due  me  ? 

42.  A  boy  bought  oranges  at  the  rate  of  3  for  5  cents, 
and  sold  them  at  the  rate  of  2  for  5  ^.  What  was  his  rate 
of  gain  ? 

43.  Ten  per  cent  of  a  number  is  32  less  than  eighteen 
per  cent  of  the  same  number.     What  is  the  number  ? 

44.  I  paid  $28,871  for  insuring  my  house  for  |3850  for 
three  years.     What  was  the  rate  of  yearly  premium  ? 

45.  A  stock  of  goods  valued  at  $6300  is  insured  for  | 
its  value  at  |%.  What  will  be  the  owner's  loss  if  the 
goods  are  totally  destroyed  by  fire? 

46.  A  man's  income  is  $  1720,  which  is  16|%  of  the  sum 
he  has  invested.     What  sum  has  he  invested  ? 

47.  From  a  cargo  containing  wheat,  1620  bu.,  or  7%,  was 
washed  overboard.     What  number  of  bushels  remained  ? 

48.  A  stock -dealer  sold  38  head  of  cattle,  which  was  4% 
of  his  entire  herd.     How  many  had  he  left  ? 

49.  In  an  orchard  containing  820  trees,  20%  of  them 
were  pear-trees,  and  the  remainder  were  plum-trees.  How 
many  plum-trees  were  there  in  the  orchard  ? 

50.  From  a  cask  of  wine  containing  65  gal.  all  but  15% 
was  sold.     How  many  gallons  were  sold? 

51.  In  a  school  containing  875  pupils  32%  of  them  are 
boys  and  the  remainder  girls.     How  many  girls  are  there  ? 


MISCELLANEOUS   REVIEW.  243 

52.  There  is  a  loss  of  $  500  on  a  house  and  lot  sold  for 
$  5000.     What  is  the  per  cent  of  loss  ? 

53.  An  agent  reports  that  he  invested  the  money  re- 
mitted him  in  wheat,  which  he  sold  at  an  advance  of  15% ; 
then  investing  the  proceeds  in  a  second  quantity,  he  was 
forced  to  sell  at  a  loss  of  12i%.  He  now  deducts  ^100 
for  expenses  and  commission,  and  remits  ^5333.75  to  his 
employer  as  the  balance  due  him.  Find  the  loss  to  the 
employer. 

54.  $90  are  paid  as  premium  for  insuring  a  block  for 
three-fourths  of  its  value.  If  the  rate  of  insurance  is  f  %, 
what  is  the  value  of  the  property  ? 

55.  In  1896  New  York  State  had  a  population  of  5,998,000, 
and  New  York  City  had  1,515,000.  What  per  cent  of  the 
population  of  the  State  lived  in  New  York  City  ? 

56.  During  the  war  of  1861-1865,  the  State  of  New 
York  paid  $  40,000,000  in  bounties  to  her  volunteers.  Her 
population  at  that  time  was  (in  round  numbers)  4,000,000. 
What  was  the  average  cost  to  each  inhabitant  ? 

57.  If  I  sell  6  horses  for  what  8  horses  cost,  what  is  my 
rate  of  gain  ? 

58.  Sold  wheat  for  $  73.54i,  by  which  a  gain  of  15%  was 
made.  What  did  the  wheat  cost,  and  what  sum  was 
gained  ? 

59.  In  a  school  containing  1160  pupils,  638  are  girls  and 
the  remainder  are  boys.     What  per  cent  are  boys  ? 

60.  A  hall  is  42  ft.  wide,  and  294  ft.  long.  What  per 
cent  of  the  length  is  the  width  ? 

61.  In  a  certain  battle  22|%  more  than  i  of  the  soldiers 
were  killed.  If  the  loss  was  110  men,  what  was  the 
original  number? 


244  PERCENTAGE. 


62.  In  an  orange-grove  S^%  of  the  trees  were  ruined 
by  frost.  If  1100  remained  uninjured,  how  many  were 
destroyed  ? 

63.  A  merchant  sold  a  lot  of  goods  for  ^  550,  thereby 
gaining  10%.     Find  the  cost  of  the  goods. 

64.  A  man  sold  a  watch  for  $32,  thereby  losing  20%  on 
the  cost.     Find  the  cost. 

65.  If  a  man  owns  66|  per  cent  of  a  factory,  and  sells 
33-J  per  cent  of  his  share  for  $  1800,  what  is  the  value  of 
the  factory  ? 

66.  A  sold  30%  of  his  steamship  to  B;  B  sold  60%  of 
his  purchase  to  C ;  C  sold  75%  of  his  share  to  D  for 
$  27,000.  What  was  the  value  of  the  vessel  ?  What  was 
each  one's  share  in  dollars  after  the  sales  had  been  made  ? 

67.  After  a  discount  of  30%  had  been  made  upon  the 
catalogue  price  of  a  book,  it  was  sold  for  $  1.75.  What 
was  the  catalogue  price  ? 

68.  Bought  a  horse  for  $  120,  and  sold  him  for  $  135. 
What  part  of  the  cost  was  the  gain?     What  per  cent? 

69.  Bought  tea  at  60  ct.  a  pound.  What  must  I  ask 
per  lb.  so  as  to  abate  10%  and  still  make  a  profit  of  25%  ? 

70.  A  merchant's  profits  for  1895  were  13402.84.  If 
they  were  6|%  less  than  in  1894,  what  were  they  in  1894  ? 

71.  In  one  week  John  solved  75  problems,  correctly. 
If  he  failed  in  16|%  of  the  number  attempted,  how  many 
were  there  in  all  ? 


SIMPLE   INTEREST. 


341.  1.  I  borrow  $  500  for  1  year,  and  at  the  end  of  the 
year  I  repay  the  money  and  6%  for  the  use  of  it.  How 
much  do  I  pay  for  the  use  of  $  500  ? 

2.  How  much  must  be  paid    for  the  use  of    $50  for  1 


10  ' 


year  at  5%  ?     At  7 

3.  How  much  at  5%  per  annum  must  I  pay  for  the  use 
of  $  1000  for  1  year  ?     For  3  years  ? 

4.  I  loan  James  Barnes  $500  at  6%.  At  the  end  of  2 
years  he  pays  me  in  full.     How  much  does  he  pay  me  ? 

Money  that  is  paid  for  the  use  of  money  is  called  Inter- 
est. The  money  for  the  use  of  which  interest  is  paid  is 
called  the  Principal,  and  the  sum  of  the  principal  and 
interest  is  called  the  Amount. 

Interest  at  6%  means  6%  of  the  principal  for  1  year. 

12  months  of  30  days  each  are  usually  regarded  as  a 
year  in  computing  interest. 

Oral. 

6.   What  is  the  interest  of  $  100  for  3  years  at  6%  ? 

Solution.  —  $  100  Principal. 
.06  Rate. 
$  6.00  Interest  for  1  year. 
3 


018.00  Interest  for  3  years. 
245 


246  SIMPLE   INTEREST. 

6.  What  is  the  interest  of  f  80  at  5%  for  2^  years  r 

7.  What  is  the  interest  of  $  1000  at  5%  for  2  yr.  6  mo.  ? 

8.  What  is  the  interest  of  $100  at  6%  for  1  year? 
For  1-1-  ?  For  2  yr.  6  mo.  ?  For  3  yr.  3  mo.  ?  For  1  yr. 
6  mo.  ? 

When  the  time  does  not  include  days,  find  interest  as 
follows : 

Principal  x  Eate  x  Time  =  Interest. 

9.  What  is  the  interest  of  $297.62  for  5  yr.  3  mo.  at 


Solution.—    $297.62 

m 

$17.8572 


Note.  — Final  results  should  not 
r  1  include  mills.    Mills  are  disregarded 

4464S  ^^  ^^^^  ^^^^^  ^'  ^^^  called  another 

892860  ^^°^  '^  ^  ^^  "^^^^• 

$93.75    Ans.. 

Find  the  interest  of : 

10.  $384.62  at  6%  for  2  yr.     12.    $  250.50  at  8%  for  5  yr. 

11.  1 463.75  at  7 %  for  3  yr.     13.    $  685.20  at  4%  for  6  yr. 

14.  $  596.15  at  5%  for  2  yr.  3  mo. 

15.  $  386.42  at  5^%  for  6  yr.  5  mo. 

16.  $  950.16  at  10%  for  41  yr. 

17.  $  283.25  at  6%  for  2  yr.  8  mo. 

Find  the  amount  of : 

18.  $  284.10  for  3  yr.  2  mo.  at  7%. 

19.  $  364.24  for  1  yr.  1  mo.  at  6%. 

20.  $  282.50  for  5  yr.  9  mo.  at  5i%. 

21.  $  298  for  4  yr.  3  mo.  at  6%. 

22.  $  389  for  7  yr.  10  mo.  at  5%. 


THE   SIX   PER   CENT   METHOD.  247 

23.  $  894  for  5^  yr.  at  5^%. 

24.  A  man  buys  a  house  and  lot  for  $  2800.  He  pays  -f 
of  the  amount  in  cash,  and  the  remainder  after  1  yr.  4  mo. 
with  5%  interest.     Find  the  amount  of  the  second  payment. 

25.  Required  the  simple  interest  and  amount  of  ^787.875 
for  7  yr.  7  mo.  at  7%. 

26.  Find  the  interest  on  a  note  for  $  12500  for  three 
months  at  8%. 

27.  A  man  paid  his  city  tax  five  months  after  it  became 
due.  His  tax  was  $  560.  In  accordance  with  city  ordi- 
nance, 1%  is  added  for  each  i  month  the  taxes  are  over- 
due. He  pays  to  the  city  collector  of  taxes,  who  adds  5% 
collection  fee.     How  much  did  he  have  to  pay  ? 

THE  SIX  PER  CENT  METHOD. 

342.  By  the  6%  method  it  is  convenient  to  find  first  the 
interest  of  $  1,  then  multiply  it  by  the  principal. 

1.  If  $.09  is  the  interest  of  $1  for  a  certain  time,  what 
is  the  interest  of  $  2  for  the  same  time  ?  of  f  10  ?  of  |  25  ? 

2.  The  interest  of  $  1  at  6%  for  a  certain  time  is  $  .034. 
What  is  the  interest  of  $  36.25  for  the  same  time  ? 

Explanation.  —  The  interest  of  $36.25  is  S6^^-q  times  the  interest 
of  $1. 

At  6%  the  interest  of  1 1  for  1  year     =     .     .     .     .      |  .06 
for  1  month  =  J^  of  $  .06    =  $  .00|. 
for  1  day       =  j\  of  f  .00^  =  $  .000^. 

3.  What  is  the  interest  of  $50.24  at  6%  for  2  yr.  8  mo. 
18  da.  ? 

Solution.  — 

The  interest  of  $  1  for  2  yr.     =    2  x  $  .06       =  $  .12 
for  8  mo.    =    8  x  $.00^     =     .04 
for  18  da.  =  18  x  $  .000^  =     .003 
The  interest  of  $  1  for  2  yr.  8  mo.  18  da.  =  $  .163 

The  interest  of  $  50.24  is  50.24  times  $ ,  163     =  $  8. 19 


248  SIMPLE  INTEREST. 

4.    What  is  the  interest  of  $  1  for  2  months?     For  6 
days  ? 

\jRule. — Find  the  interest  on  $  1  for  the  given  time,  and 
multiply  it  by  the  priyicipal,  considered  as  an  abstract 
number. 
Or,  multiply  the  number  of  dollars  by  the  number  of  days, 
and  divide  by  6.  The  quotient  will  be  the  interest  in 
mills. 

rind  the  interest  at  6%  of : 

5.  $  382  for  6  mo.  24  da. 

6.  $  58.63  for  1  yr.  5  mo.  17  da. 

7.  f  256  for  3yr.  5  mo. 

8.  f  249.83  for  1  yr.  2  mo.  15  da. 

9.  $  51  for  236  da. 

10.  $  74  for  2  mo.  19  da. 

11.  f  1500  f or  1  yr.  3  da. 

12.  f  287.15  for  2  yr.  11  mo.  22  da. 

Interest  at  any  rate  per  cent  may  be  found  as  follows : 
At  7%,  find  interest  at  6%,  and  add  -|-  of  itself. 
At  5%,  find  interest  at  6%,  and  subtract  \  of  itself. 
At  8%,  find  interest  at  6%,  and  add  f  or  ^  of  itself. 
At  4%,  find  interest  at  6%,  and  subtract  f  or  ^  of  itself 
At  41%,  find  interest  at  6%,  and  subtract  \  of  itself. 

Find  the  interest  and  amount  of  the  following : 

13.  8  2350  for  1  yr.  3  mo.  6  da.  at  5%. 

14.  ,  $  125.75  for  2  mo.  18  da.  at  7%. 

15.  $  950.63  for  3  yr.  17  da.  at  41%. 

16.  ^  336.48  for  90  da.  at  71%. 


THE   SIX  PER   CENT   METHOD.  249 

17.  $  738.53  for  2  yr.  2  mo.  24  da.  at  8%. 

18.  $  5000  for  6  mo.  19  da.  at  4%. 

19.  $  867.35  for  1  yr.  3  mo.  27  da.  at  9%. 

20.  $  260.50  for  5  yr.  21  da.  at  10%. 

21.  $  3050  for  3  yr.  3  mo.  3  da.  at  12%. 

22.  $  625.57  for  1  yr.  2  mo.  15  da.  at  3%. 

23.  A  grocer's  bill  for  $  84.36  is  paid  8  mo.  12  da.  after 
it  becomes  due,  with  interest  at  5%.     How  much  is  paid  ? 

24.  Find  the  interest  at  7%  on  $  37200  for  5  days. 

25.  A  note  for  $  125  was  dated  March  1,  1894.  What 
was  due  Aug.  5,  1895,  int.  at  6%  ? 

26.  Find  the  amount  of  $  460.50  for  2  yr.  7  mo.  15  da.  at 
5%. 

27.  What  is  the  amount  of  a  note  for  $  360  due  in  3  mo., 
interest  at  5%  ? 

343.  On  short-time  notes,  it  is  customary  to  compute 
interest  for  the  actual  number  of  days,  using  the  6% 
method. 

Find  the  amount  of : 

28.  I  684.23  from  June  5,  1895,  to  July  23,  1895,  at  6%. 

29.  1 846  from  Jan.  6  to  March  9,  1896,  at  5%. 

30.  $  2064.28  from  April  13, 1894,  to  June  3, 1894,  at  8%. 

31.  $  1428  from  May  12,  1892,  to  June  9,  1892,  at  6%. 

32.  $  324  from  April  1,  1896,  to  June  4,  1896,  at  7%. 

33.  $  3500  from  Feb.  9, 1895,  to  March  12,  1896,  at  4i%. 

34.  $  862.15  from  May  25,  1893,  to  July  22,  1893,  at  6%. 

35.  What  is  the  amount  of  a  note  of  $384.16  at  6%, 
given  June  11,  1896,  and  paid  Aug.  12,  1896  ? 


250  SIMPLE   INTEREST. 

36.  A  note  of  $395.80  dated  April  5,  1896,  was  paid 
Aug.  4,  1896.     What  was  the  amount  ? 

37.  On  Dec.  9,  1894,  John  Smith  borrowed  $  484,  agree- 
ing to  pay  interest  at  o%.  He  paid  the  debt  in  full  on 
March  3,  1895.     What  did  he  pay  ? 

38.  What  is  the  amount  of  $  58.24  at  7%  from  April  23, 
1893,  to  July  22,  1893  ? 

39.  A  bill  of  $312  with  interest  at  5%  was  paid  at  the 
end  of  90  days.     What  was  the  amount  ? 

40.  What  is  the  interest  of  $  30000  at  8%  for  7  days  ? 
344.    rind  the  interest,  using  the  best  method. 


PRINCIPAL. 

TIME. 

RATE. 

41. 

$364, 

3yr., 

8%. 

42. 

$  692.15, 

1  yr.  3  mo., 

9%. 

43. 

$342, 

62  da., 

6%. 

44. 

$  243.50, 

2  yr.  5  mo.  18  da.. 

7%. 

45. 

$  .392, 

1  yr.  3  mo.  15  da., 

4%. 

46. 

$  150.16, 

7  yr.  2  mo.  27  da.. 

4^95.. 

47. 

$  284.10, 

1  yr.  8  mo.  18  da.. 

6%. 

48. 

$  1400, 

2  yr.  1  mo.  12  da., 

7%. 

49. 

$124, 

5  yr.  3  mo.  29  da., 

6%. 

50. 

$48, 

33  da.. 

.     6%. 

51. 

$124, 

112  da.. 

5%. 

52. 

$315, 

45  da.. 

^fo. 

53. 

$214, 

93  da., 

8%. 

teXACT   INTEREST.  251 

345.  Find  the  amount  of : 

54.  $365  from  April  1,  1895,  to  July  5,  1897,  at  6%. 

55.  $250  from  July  3,  1891,  to  April  21,  1893,  at  9%. 

56.  f  582  from  Sept.  4,  1896,  to  July  8,  1897,  at  8%. 

57.  $346.18  from  May  10,  1893,  to  March  10,  1895,  at 
6%. 

58.  $287  from  Jan.  1,  1895,  to  July  1,  1897,  at  4^%. 
69.    $  1684  from  July  17,  1896,  to  Sept.  5,  1898,  at  1\%. 

60.  $2500  from  April  16,  1873,  to  Oct.  11,  1881,  at  5%. 

61.  $  186  from  Feb.  12,  1896,  to  March  4,  1896,  at  6%. 

62.  $346  from  March  11,  1895,  to  Feb.  11,  1896,  at  6%. 

EXACT  INTEREST. 

346.  When  the  time  includes  days,  interest  computed  by 
the  6%  method  is  not  strictly  exact,  by  reason  of  using 
only  30  days  for  a  month,  which  makes  the  year  only  360 
days.  The  day  is  therefore  reckoned  as  -^\-^  of  a  year, 
whereas  it  is  -^\-^  of  a  year. 

Rule.  —  To  compute  exact  interest,  find  the  exact  time  in 
days,  and  consider  1  day's  interest  as  -g-i-g-  of  1  year's 
interest. 

1.    Find  the  exact  interest  of  $358  for  74  days  at  7%. 

Solution. —  $  358  x  .07  =  $25.06,  1  year's  interest.  74  days'  in- 
terest is  ^-^  of  1  year's  interest.     -^  of  $  25.06  =  $  5.08.     Ans. 

Find  the  exact  interest  of : 

2.  $324  for  15  d.  at  9%. 

3.  $253  for  98  d.  at  4%. 

4.  $624  for  117  d.  at  7%. 

5.  $  153.26  for  256  d.  at  ^%. 

6.  $  620  from  Aug.  15  to  Nov.  12  at  6%. 


252  SIMPLE  INTEREST. 

7.  f  540.25  from  June  12  to  Sept.  14.  at  8%. 

8.  $7560  for  90  days  at  5i%. 

9.  Find  the  exact  interest  at  5%  on  a  note  dated  Jan. 
14,  1896,  and  paid  March  31,  1896,  for  $832. 

10.  Find  the  exact  interest  on  $  800  for  219  days  at  41%. 

11.  A  city  treasurer  deposits  $387,913.56  in  the  banks 
at  2%  per  annum.  What  interest  will  the  city  receive  in 
5  days  ? 

12.  On  June  4,  1895,  a  coal-dealer  bought  of  the  D.  L. 
&W.  R.  R.  235  tons  of  chestnut  coal  at  $4.10  per  ton. 
At  6%  what  will  be  the  exact  interest  on  the  amount  on 
Jan.  1,  1896  ? 

PROBLEMS  IN  INTEREST. 

347.  To  find  the  Rate,  when  Principal,  Interest,  and  Time 
are  given. 

1.  What  is  the  rate  when  the  interest  of  $  250  for  4 
years  is  $60? 

$10^S60  Solution.  — The  interest  on  the 

6  timP^  1  ^   -ao/.      principal  at  1%  for  4  years  =  flO. 

/^  —  ^  /o      gi„(je  |iQ  ig  ^i^g  interest  at  1  %,  f  60 

must  be  the  interest  at  as  many  times  1  %  as  $  10  is  contained  times  in 

$  60,  which  are  6  times.     Therefore  the  rate  is  6  times  1  %  =  6  %. 

^      Rule.  —  Divide   the    given   interest    by    the    interest  of   the 
principal  for  the  given  time  at  1%. 

2.  A  man  borrowed  $4625  for  5  yr.  8  mo.  18  da.,  and 
paid  $  1586. 37|-  for  the  use  of  it.  What  was  the  rate  of 
interest  ? 

3.  If  $30.40  is  paid  for  the  use  of  $960  for  7  mo.  18 
da.,  what  is  the  rate  per  cent  ? 

4.  At  what  rate  per  cent  must  $1450  be  loaned  for 
4  yr.  5  mo.  to  yield  $  576.37^  ? 


PROBLEMS   IN   INTEREST.  253 

5.  At  what  rate  will  $  1730  amount  to  ^2048.32  in  4  yr. 
7  mo.  6  da.  ? 

6.  4  yr,  7  mo.  6  da.  after  its  date  a  note  for  ^1730 
amounted  to  $  2048.32.     What  was  the  rate  of  interest  ? 

7.  At  what  rate  %  must  $  5600  be  invested  for  1  yr.  4 
mo.  to  bear  $560  interest  ? 

8.  A  Kansas  farmer  has  a  mortgage  on  his  farm  for 
$  1250.  What  rate  of  interest  does  he  pay,  if  the  interest 
for  2  yr.  6  mo.  equals  ^  of  the  debt  ? 

9.  At  what  rate  must  $2800  be  invested  to  yield  a 
semi-annual  interest  of  $  112  ? 

10.  At  what  rate  will  $600  yield  $  198  in  5  years  and  6 
months  ? 

11.  At  what  rate  will  any  sum  double  itself  in  20  yr.  ? 

348.  To  find  Time,  when  Principal,  Interest,  and  Rate  are 
given. 

1.  In  what  time  will  $250  gain  $  60  at  6%  ? 

Solution.  —The  interest  of  |250  for  1  year  at  6%  =  $  15.  Since 
$  15  is  the  interest  for  1  year,  $  60  is  the  interest  for  as  many  years 
as  $  15  is  contained  times  in  f  60  =  4  years. 

Rule. — Divide  the    given    interest   by  the  iyiterest    of   the 
principal  for  1  year. 

2.  In  what  time  will  $  600  yield  $  91.50,  interest  at  6% ? 

$  600  X  .06  =  $  36,  interest  on  the  principal  for  1  year. 

$91.50 -^  36  =  2.5416  +  years.      Reducing  the  decimal  part  of 

the  time  to  months  and  days,  we  have  6  mo.  15  da. 
The  answer  is  2  yr.  6  mo.  15  da. 
Note.  —  A  decimal  less  than  .5  of  a  day  is  not  counted,  but  .5  or 
more  is  counted  another  day. 

3.  In  what  time  will  $530  gain  $92.75  interest  at  5%  ? 

4.  In  what  time  will  $  400  yield  $  ^5  interest  at  51%  ? 


254  SIMPLE   INTEREST. 

5.  In  what  time  will  $500  gain  $15  at  6%  ? 

6.  In  what  time  will  $4625  yield  $1586.38  at  6%  ? 

7.  In   what  time  will  $1730  amount  to  $2048.32  at 

4%  ? 

8.  The  face  of  a  note  was  $960,  rate  of  interest  5%, 
and  the  interest  $  30.44.     How  long  did  it  run  ? 

9.  I   borrowed   $1284   at   41%,   and   kept   it   until   it 
amounted  to  $  1421.067.     How  long  did  I  keep  it  ?     • 

10.  For  how  long  will  $  2700  have  to  be  invested  to 
amount  to  $2976.25  at  5%  ? 

11.  A  man  received  $9.73  interest  on  $556  at  7%. 
What  was  the  time  ? 

12.  In  what  time  will  any  sum  double  itself  at  6%  ? 

349.  To  find  Principal,  when  Interest  or  Amount,  Rate,  and 
Time  are  given. 

1.  What  principal  at  6%  will  gain  $60  interest  in  4 
years  ? 

$.24)$  60.00(250. 

4g  Solution.  —  Since  1  dollar  in  4  years  will 

T^  gain   $.24   interest,    it   will   take    as    many 

^9/.  dollars  to  gain  .^60  interest  as  $.24  is  con- 

— —  tained  times  in  $60,  or  $250. 

^    Rule.  —  Divide  the  given  interest  hy  the  interest  of  $  1  for  the 
given  time  and  rate. 

2.  What  principal  at  6%   will  amount  to  $310  in  4 
years  ? 

Solution.  —  Since  $1.24  is  the  amount  of  $1  for  4  years,  $.310 
must  be  the  amount  of  as  many  times  $  1  as  $1.24  is  contained  times 
in  310  =  $250. 

3.  What  sum  invested  at  5%  will  give  a  yearly  income 
of  $500? 


J 


PROMISSORY  NOTES.  255 

4.  What  principal  will  yield  $25  in  6  mo.  at  5%  ? 

5.  What  principal   in  3  yr.   6    mo.   at   5%    will   yield 
$  92.75  interest  ? 

6.  What  sum  of  money  will  produce  $  1586.37^  in  5  yr. 

8  mo.  18  da.  at  6%  ? 

7.  What  principal  will  yield  $318.32  in  4  yr.  7  mo. 
6  da.  at  4%  ? 

8.  What  principal  will  pay  $  1556.77-5  interest  in  2  yr. 

9  mo.  at4i%? 

9.  The  amount  is  $1093.921   time  2  yr.  3  mo.  27  da., 
rate  5%.     What  is  the  principal  ? 

10.    It  required  $407.65  to  pay  a  loan  at  8%  for  7  mo. 
24  da.     What  sum  was  loaned  ? 


PROMISSORY  NOTES. 

350.  A  Promissory  Note  is  a  written  promise  to  pay  a 
sum  of  money  at  a  certain  time. 

,  351.  At  least  two  parties  must  be  named  in  the  note, 
the  Maker  and  the  Payee. 

The  Maker  makes  the  promise  to  pay  to  the  Payee  the 
sum  named  in  the  note.  This  sum  is  called  the  Face.  The 
owner  of  a  note  is  called  the  Holder. 

Each  State  has  a  lawful  or  legal  rate  of  interest. 

If  no  rate  is  fixed  in  the  note,  the  legal  rate  is  under- 
stood.    (See  Appendix.) 

352.  Interest  higher  than  the  legal  rate  is  Usury. 

353.  A  note  is  Negotiable  when  payable  to  the  bearer,  or 
to  the  order  of  the  payee.  It  is  called  negotiable  because 
it  can  be  negotiated ;  i.e.  bought  and  sold. 


256  SIMPLE   INTEREST. 

354.    The  two  forms  of  notes  given  below  are  negotiable : 

NOTE  1. 

$510^  Chicago,  111.,  Jan.  8,  1901 

Two  months  after  date,  I  promise  to  pay  to  the 
order  of Charles  M.  Warner Five  hun- 
dred  ten    and   ~  Dollars,   for   value   received,   at   the 

First  National  Bank. 

U.  B.   Smith, 


$216^  Des  Moines,  la.,  Juli/  10,  1901 

One  ifear  after  date,  for  value  received,  I  promise 

to  pay  Asa  D.   Tucker  or  bearer,  Two  hundred  sixteen 

and  -—  Dollars,  with  interest. 

John  T.  Brown. 

Note. — A  note  may  be  payable  on  a  given  day  ;  as,  On  March  15 
after  date,  I  promise  to  pay,  etc.  A  note  may  be  payable  on  demand ; 
as.  On  demand  I  promise  to  pay,  etc. 

355.  A  note  made  payable  to  the  payee  only  is  called 
a  non-negotiable  note. 

Note.  —  When  a  note  is  payable  in  a  State  in  which  three  days  of 
grace  are  allowed,  maturity  is  three  days  after  the  expiration  of  the 
interval  named  in  the  note. 

356.  1.    Write  a  negotiable  note,  bearing  interest. 
Indorse  it  with  payee's  name,  and   find  the  amount  at 

maturity. 

To  whom  must  the  maker  pay  the  money  ? 


PKOMISSORY  KOTES.  257 

2.  Write  a  non-negotiable  note  payable  on  a  specified 
date,  and  find  the  amount  due  at  maturity, 

3.  Write  a  negotiable  note,  and  find  the  amount  of  it. 

4.  Write  a  non-interest-bearing  demand  note. 

5.  Write  a  non-negotiable  interest-bearing  note. 

6.  Write  a  6%  note,  dated  June  15,  1901,  payable  in  1 
year  without  interest,  with  yourself  as  payee,  and  your 
teacher  as  maker,  and  find  the  amount  of  it  to  the  present 
time. 

7.  Write  a  negotiable  note,  using  the  following : 

Date,  Jan.  16,  1894;  Time,  6  months;  Face,  $1684.96; 
Payee,  Andrew  Jackson;  Maker,  Silas  Wright;  Interest 
at  6%.  Indorse  it,  showing  that  the  maker  has  transferred 
it  to  another.  What  is  the  amount  of  the  note,  if  paid  in 
full  Nov.  11,  1894  ? 

357.  A  note  should  contain : 

1.  The  face  in  figures  at  the  left  upper  corner. 

2.  The  place  and  date  at  the  right  upper  corner. 

3.  The  time  of  payment. 

4.  The  words  "Value  Received." 

5.  The  face  written  in  words  in  the  body  of  the  note. 

6.  The  place  at  which  it  is  payable. 

7.  The  words  "  with  interest,"  if  agreed  upon. 

358.  A  note  is  said  to  mature  on  the  day  on  which  it  is  due. 

359.  A  note  that  does  not  contain  the  words  "  with  inter- 
est "  bears  interest  from  maturity,  if  not  paid  at  that  time. 

When  does  interest  begin  in  Note  1  ?     In  Note  2  ? 

360.  In  many  of  the  States  the  maker  is  allowed  three 
days  (called  Days  of  Grace)  in  which  to  pay  a  note,  after 
the  time  named  in  the  note  has  expired.  In  these  States, 
the  date  of  maturity  falls  on  the  last  day  of  grace. 


258  SIMPLE   INTEREST. 

Days  of  grace  are  not  allowed  in  California,  Connecticut, 
District  of  Columbia,  Idaho,  Illinois,  Maryland,  Massachu- 
setts, Montana,  New  Jersey,  New  York,  North  Dakota, 
Ohio,  Oregon,  Pennsylvania,  Utah,  Vermont,  and  Wisconsin. 

Note.  —  When  the  holder  of  a  note  transfers  it  to  another,  he  is 
usually  required  to  indorse  it,  i.e.  to  write  his  name  across  the  back. 
This  is  required  as  an  order  to  the  maker  to  pay  the  money,  when 
due,  to  the  new  holder.  An  indorser  is  also  resiDonsible  for  the  pay- 
ment of  a  note  in  case  the  maker  fails  to  pay  it  when  due. 

PARTIAL   PAYMENTS. 

361.  Payments  in  part  of  a  note  or  other  debt  are 
Partial  Payments. 

The  Supreme  Court  of  the  United  States  has  adopted 
the  following  rule  for  finding  the  amount  due  on  a  note 
after  partial  payments  have  been  made. 

UNITED    STATES    RULE. 

Find  the  amount  of  the  principal  to  the  time  when  the  ])ay- 
ment   or   sum   of  the  payments   equals   or   exceeds    the 
interest  then  due. 
Deduct  from  this  amount  the  payment  or  payments. 
Treat  the  remainder  as  a  new  principal^  and  so  proceed 
until  the  date  of  settlement. 

Note.  —  When  a  partial  payment  of  a  note  or  other  contract  is 
made,  the  holder  writes  upon  the  back  of  it  the  sum  paid,  with  the 
date  of  payment.  Sums  so  written  are  called  indorsements.  The 
common  form  of  indorsement  is  as  follows  • 

Received  on  the  within, 

July  16,  1896,     $ 

1.  A  note  for  $1820  was  given  Jan.  1,  1892,  and  settled 
July  13,  1894.  The  following  payments  were  indorsed 
upon  it:  May  25,  1892,  $250;  Jan.  25,1893,  $45;  April  7, 
1893,  $  375 ;  July  13,  1893,  $  750.  How  much  was  due  on 
the  day  of  settlement,  interest  at  6%  ? 


PARTIAL   PAYMENTS. 


259 


First  write  the  note,  and  properly  indorse  the  payments 
upon  the  back  of  it. 


YB.       MO. 

I>A. 

PAYME>TS. 

1892      5 
1892      1 

25 
1 

$250 

$1820  Principal. 
.024 

4 

24 

$43.68 
1820.00 

.024 

$1863.68  1st  Amount. 
250.00  1st  Payment. 

Ite      1 
189\    5/ 

4 

$45. 

^lr61^Ua^i>[ew  Principal.^ — ■ — ^ 

JK 

$64,..5&-tntefest  exceBds-Ea^yment. 

1893      4 
1892      5 

7 
25 

$375 

$1613.68 
.052 

10 

12 

$83.91 
1613.68 

.052 

$420 

$1697.59  Amount. 

420.00  Sum  of  2d  and  3d  Payments. 

1893      7 
1893      4 

13 

7 

$750 

$1277.59  New  Principal. 
.016 

3 

6 

20.44 
1277.59 

.016 

$1298.03  Amount. 
750.00  4tli  Payment. 

1894      7 
1893      7 

13 
13 

Settled. 

$548.03  New  Principal. 
.06 

1      0 

0 

$28.88 
548.03 

.06 

$576.91     Ans. 

Note. — The  $45  payment,  being  less  than  the  interest  ($64.55),' 
is  not  deducted  from  the  amount  of  the  second  principal  ($1678,23). 
If  this  were  done,  and  the  remainder  treated  as  a  new  principal,  a 
portion  of  it  ($19.55),  being  interest,  would  draw  interest,  which  is 
not  legal.  Therefore,  interest  must  be  taken  on  $1613.68  until  the 
date  of  the  next  payment  (10  mo.  12  da.).  The  sum  of  the  two 
payments,  being  greater  than  the  interest,  is  subtracted  from  the 
amount. 


260  SIMPLE  INTEREST. 

Write  in  proper  form  on  paper  a  note  for  each  of  the 
following,  indorse  the  payments,  and  solve: 

2.  Date,  Jan.  1,  1874,  at  New  Orleans,  La.  Face,  $1000. 
Interest  at  6%.  Indorsements:  July  7,  1874,  $400;  Oct. 
19,  1874,  $300;  Dec.  1,  1874,  $100.  What  remains  due 
Jan.  1,  1875? 

3.  Face,  $900.  Date,  March  1,  1886.  Interest  at  9%. 
Indorsements:  Aug.  10,  1886,  $300;  Sept.  1,  1886,  $100; 
Jan.  1,  1887,  $50.     What  was  due  March  1,  1887  ? 

4.  Face,  $2000.  Date,  Jan.  20,  1892.  Interest  at  6%. 
Indorsements:  May  20,  1892,  $100;  July  20,  1893,  $100; 
Sept.  10,  1893,  $700;  Oet.  20,  1894,  $75.  Settled  Oct.  20, 
1895.     What  was  due  ? 

6. 

^300  Binghamton,  N.Y.,  Oct.  12,  1889 

On  demand,  for  value  received,  /promise  to  pay 
S.  D.  Cleveland^^^^^.^^,^.^^^,^,^,^^ov  order,  Three  hun- 
dred Dollars,  with  interest. 

J.  H,  Van  Alstyne. 

The  following  payments  were  made  on  this  note:  June 
27,  1891,  one  hundred  fifty  dollars;  Dec.  9,  1892,  one 
hundred  fifty  dollars.     What  was  due  Oct.  9,  1895? 

6.  On. a  note  for  $573.25,  at  6%,  dated  June  10,  1888, 
were  the  following  indorsements  :  May  20,  1889,  $50;  July 
10,1890,  $16.50;  April  5,  1891,  $14.30;  July  14,  1892, 
$250.     How  much  was  due  Sept.  20,  1893? 

7.  A  note  of  $850  was  dated  June  21,  1892,  bearing 
interest  at  6%.  On  this  note  were  the  following  indorse- 
ments:  Sept.  15,  1892,  $150.90;  Nov.  21,  1893,  $45;  Jan. 
15, 1894,  $256.88.     What  remained  due  June  21,  1894  ? 


merchants'  rule.  261 

8.  Find  what  was  due  June  1,  1896,  on  a  note  for 
$1928,  with  41%  interest,  dated  Jan.  1,  1891,  and  bearing 
the  following  indorsements:  March  1,  1891,  $300;  Oct.  16, 
1893,  $40;  Feb.  4,  1894,  $800;  Dec.  16,  1895,  $500. 

9.  On  a  note  for  $832.26  dated  Aug.  3,  1889,  due  in 
6  months,  the  following  payments  were  indorsed:  $350, 
Oct.  5,  1890;  and  $468.37,  May  15,  1892.  How  much  was 
due  Dec.  12,  1893,  interest  at  7%  ? 

10.  Face,  $2950.  Date,  July  1,  1885.  Interest,  7%. 
Indorsements:  Oct.  1,  1885,  $750;  Jan.  15,  1886,  $600; 
July  1,  1886,  $900;  Dec.  1,  1886,  $300;  March  1,  1887, 
$450.     What  was  due  July  1,  1887? 

MERCHANTS'   BULB. 

362.  When  notes  and  accounts  are  settled  within  a  year 
after  interest  begins,  and  upon  which  partial  payments 
have  been  made,  it  is  customary  for  business  men  to  make 
use  of  the  following  rule : 

Find  the  amount  of  the  entire  debt  at  date  of  settlement. 
Find  the  amount  of  each  payment  at  date  of  settlement. 
Subtract  the  amount  of  the  payments  from  the  amount  of 
the  debt. 


$648^  Dubuque,  la.,  Ma^  i,  1896 

For   value   received,   /  promise    to   pay^ ^ 

D.   McCarthy/   ^    Co.., or  bearer.   Six  hundred 

forty-eight  Dollars  on  demand,  with  interest. 

Charles  U.    White. 


262  COMPOUND    INTEREST. 

Indorsements:  June  1,  1896,  f  JoO;  Aug.  1,  1896,  $200; 
Oct.  1,  1896,  $300.     Interest  at  6%. 
What  was  due  Dec.  1,  1896  ? 

Solution, — $648  in  7  mo.  amounts  to  $670.68 

$150  in  6  mo.  amounts  to  $154.50 
$200  in  4  mo.  amounts  to  $204.00 
$300  in  2  mo.  amounts  to  $303.00        661.50 

$9.18 

2.  On  a  note  of  $1186.48,  with  interest  at  5%,  dated 
April  4,  1890,  these  payments  were  indorsed:  July  10, 
1890,  $250;  Aug.  4,  1890,  $300;  Dec.  8,  1890,  $150;  Jan. 
2,  1891,  $  75.     How  much  was  due  Feb.  4,  1891  ? 

3.  On  Oct.  16,  1896,  John  D.  Wilson  gives  his  note  for 
$483.98,  with  interest  at  6%.  He  pays  the  note  in  full, 
March  28,  1897,  having  made  a  payment  of  $350  on  Jan. 
2S,  1897.     How  much  does  it  require  to  settle  the  note  ? 

COMPOUND    INTEREST. 

363.  Compound  Interest  is  interest  on  unpaid  interest,  as 
well  as  on  the  principal,  at  the  end  of  regular  interest  periods. 

Note. — Interest  is  compounded  annually,  semi-annually,  or  quar- 
terly, according  to  agreement. 

Compound  interest  is  not  authorized  by  law.  It  is  customary  for 
savings  banks  to  allow  interest  on  interest  when  it  has  been  on  deposit 
for  a  full  interest  period. 

1.  Find  the  compound  interest  of  $  350  for  2  years  and 
6  months  at  6%. 

Solution.  —  $350.00  Principal. 

21.00  Interest  for  1st  year. 
$371.00  Amount  taken  as  new  principal. 

22.26  Interest  for  2d  year. 
$393.26  Amount  used  as  new  principal. 

11.80  Interest  for  6  mo. 
$405.06  Amount  for  2  yr.  6  mo. 
350.00  1st  principal. 
$55.06  Compound  interest  for  2  yr.  6  mS. 


REVIEW   OF   INTEREST.  263 

Note.  —  When  the  interest  is  compounded  semi-annually,  the  rate 
is  one-half  the  annual  rate  for  each  period.  When  quarterly,  one- 
fourth,  etc. 

When  no  interest  period  is  mentioned,  interest  is  compounded 
annually. 

2.  What  is  the  compound  interest  of  $830  for  3  years 
at  5  per  cent  ? 

3.  What  is  the  amount  of  $650  for  4  years  at  4%  in- 
terest, compounded  semi-annually  ? 

4.  What  is  the  compound  interest  of  $365  for  2  yr.  7 
mo.  18  da.  at  6%,  compounded  semi-annually? 

5.  What  is  the  compound  interest  on  $640  for  4  years 
at  5%  ? 

6.  What  is  the  interest,  compounded  quarterly,  on 
$538.25  for  2  yr.  6  mo.,  rate  4%  ? 

7.  What  is  the  interest,  compounded  annually,  on 
$683.48  for  four  years  at  6%  ? 

8.  What  is  the  compound  interest  on  $437.50,  for  3  yr. 
6  mo.,  at  5%,  compounded  semi-annually  ? 

REVIEW   OF  INTEREST. 

364.  1.  What  is  simple  interest  ?  Compound  interest? 
A  promissory  note  ?     A  negotiable  note  ? 

2.  Define  payee,  holder,  signer  or  maker. 

3.  Describe  two  common  methods  of  computing  interest. 

4.  Prove  that,  at  6%,  6  cents  is  the  interest  on  $  1  for  1 
year. 

5.  Prove  that  5  mills  is  the  interest  gn  $  1  for  1  month. 

6.  Prove  that  ^  mill  is  the  interest  on  $  1  for  1  day. 

7.  Why  is  interest  not  accurate  when  computed  by  the 
6%  method? 


264  INTEREST. 

8.  Find  the  interest  on  $50000  for  252  days  by  the 
6%  method,  then  by  the  exact  interest  method.  Which  is 
more  favorable  to  the  payee  ? 

9.  When  does  a  note  mature  ? 

10.  What  elements  must  be  given  when  we  find  inter- 
est?    Rate?     Time?     Principal? 

11.  How  do  you  find  the  rate?  The  time?  The  prin- 
cipal ? 

12.  What  are  days  of  grace  ? 

13.  Does  the  maker  of  a  non-interest-bearing  note  ever 
have  to  pay  interest  ?     Explain. 

14.  What  use  is  made  of  compound  interest? 

15.  Find  the  compound  interest,  then  the  simple  interest, 
at  6%  on  $  25000  for  5  years,  and  note  the  difference. 

16.  When  a  note  is  not  paid  at  maturity,  why  is  it  to  the 
holder's  advantage  to  require  a  new  note  ? 

17.  What  is  the  effect  of  a  payee's  indorsement  ? 

18.  When  a  partial  payment  is  made  that  does  not  equal 
the  interest  due,  why  is  not  the  payment  subtracted  from 
the  amount  ? 

19.  Solve  a  problem  in  partial  payments  by  both  the 
United  States  and  the  merchants'  rule.  Which  is  more 
favorable  to  the  payer  ? 

20.  Find  the  compound  interest  on  $  1420.80  for  1  yr. 
9  mo.  at  6%,  computed  semi-annually. 

21.  Find  the  ameunt  of  a  debt  of  $5672.00  for  4  years 
at  4%  compound  interest. 

22.  Find  the  interest  on  $720  at  6%  for  2  yr.  8  mo. 
22  days. 


REVIEW  OF  INTEREST.  265 

Find  the  interest  on : 

23.  $  675.20  for  3  yr.  5  mo.  at  7%. 

24.  $  754.30  for  1  yr.  4  mo.  15  da.  at  51%. 

25.  $  564.11  for  2  yr.  3  mo.  18  da.  at  4%. 

26.  A  county  in  Missouri  owes  $  85,640.  In  how  many 
days  will  the  interest  at  6%  amount  to  $  897.22  ? 

27.  Find  the  interest  at  8%  on  $3960.36  for  9  mo.  20 
da. 

28.  Find  the  amount  of  $  2536.48  for  1  yr.  3  mo.  18  da. 

at  7%. 

29.  The  interest  on  $  600  for  3  yr.  6  mo.  was  $  126. 
What  was  the  rate  ? 

30.  The  interest  on  a  note  for  $460.50  at  5%  was 
$  60.44.     What  was  the  time  ? 

31.  The  interest  on  a  certain  sum  was  $  96.04,  the  rate 
6%.     Find  the  principal. 

32.  The  amount  due  on  a  6%  note  due  in  1  yr.  5  mo. 
4  da.  was  $  135.708.     What  was  the  face  of  the  note  ? 

33.  Find  the  exact  interest  on  a  note  for  $  600,  dated 
Aug.  5,  1895,  and  due  July  1,  1896,  interest  at  6%. 

34.  Find  the  amount  and  simple  interest  of  $623.74,  one 
half  of  which  is  to  be  paid  in  2  yr.  3  mo.  at  4%,  the  other 
half  to  be  paid  in  3  yr.  5  mo.  at  6%. 

35.  A  note  for  $  146.20,  dated  June  5,  1869,  was  paid 
July  11,  1872,  with  interest  at  6  per  cent.  What  was  the 
interest  ? 

36.  A  man  borrowed,  Dec.  25,  1877,  $  137.40  at  6% 
interest,  and  kept  it  until  Jan.  15,  1880.  What  was  the 
interest  ? 


266  INTEREST. 

37.  Payments  were  made  on  a  note  of  $  1800  dated  Jan. 
12,  1891,  as  follows  :  March  6, 1891,  $  300 ;  April  15,  1891, 
$  190 ;  July  3,  1891,  f  565 ;  Oct.  15,  1891,  $  700.  What 
was  due  Dec.  21, 1891,  interest  at  6%  ? 

38.  When  must  $  1600  be  put  at  interest  at  6%,  so  that 
it  will  amount  to  $1800  on  Jan.  1,  1898  ? 

39.  Find  the  amount  of  $  375  for  2  yr.  8  mo.  16  da.  at 


40.  Find  the  amount  at  simple  interest  of  $  1200  from 
April  4,  1895,  to  the  present  time. 

41.  A  note  for  $  728  is  dated  Nov.  16,  1894.  March  8, 
1895,  there  was  paid  on  it  $  25.  Find  the  amount  due  on 
Jan.  4,  1896,  interest  at  6%. 

42.  Find  the  amount  at  simple  interest  of  $  1184.63  for 
1  yr.  4  mo.  17  da.  at  ^%  ? 

43.  Write  your  own  promissory  note  for  $  200,  with 
interest,  payable  in  60  days  from  to-day.  When  does  it 
becoijie  due  ?     Find  the  amount  due  at  maturity. 

44.  Find  the  exact  interest  on  $843.20  from  April  10, 
1895,  to  March  15,  1896,  at  4^%. 

45.  Upon  a  note  for  $  950,  dated  Syracuse,  N.Y.,  Jan. 
1,  1894,  i  150  was  paid  Aug.  16,  1894 ;  $  25  March  1, 1896; 
and  i  200  April  16,  1896.     How  much  is  due  to-day  ? 

46.  .$645  was  paid  as  interest  on  $  2000  for  3  yr.  7  mo; 
What  was  the  rate  ? 

47.  $30  was  paid  as  interest  on  $600  at  6%.  What 
was  the  time  ? 

48.  A  house  that  cost  $  5000  was  rented  for  $  500,  and 
$  100  was  paid  for  annual  taxes  and  repairs.  What  rate  of 
interest  did  the  investment  yield  ? 


O^RUE  DISCOUNT.  267 

49.  A  person  investing  a  certain  sum  of  money  at  6% 
for  1  yr.  6  mo.  found  at  the  end  of  that  time  the  invest- 
ment amounted  to  $  545.     Find  the  sum  invested. 

50.  A  man  bought  a  horse  for  $  150,  paying  $  70  in  cash, 
and  the  balance  on  time  at  6%.  He  paid  at  the  time  of 
settlement  $  83.60.  How  much  time  elapsed  before  that 
date  ? 

51.  H.  C.  Harmon  loaned  $250  for  1  yr.  3  mo.  27  da., 
which  amounted  to  $  269.875  at  the  time  of  payment. 
Find  the  rate  of  interest. 

52.  A  person  having  a  certain  sum  of  money  invested, 
and  drawing  compound  interest  at  6%,  found  at  the  end  of 
2  yr.  2  mo.  that  it  amounted  to  $  567.418.  What  was  the 
sum  invested  ? 

53.  A  sum  of  money  was  borrowed  Jan.  30,  1895,  and 
$  419.60  paid  in  full  Nov.  24,  1895.  The  rate  of  interest 
being  6%,  how  much  of  this  was  interest  ? 

54.  A  man  owes  $4600  at  7%,  and  each  payment  of  in- 
terest amounts  to  $  161.     How  often  does  he  pay  interest  ? 

TRUE  DISCOUNT. 
365.   Oral. 

1.  What  will  be  the  amount  of  $100  at  6%  one  year 
from  to-day  ? 

2.  What  is  the  value  to-day  of  a  debt  of  $106,  due  in 
one  year,  when  money  is  worth  6%  interest? 

3.  How  much  money  paid  to-day  will  cancel  a  debt  of 
$112,  due  two  years  hence,  money  being  worth  6%? 

4.  What  is  the  present  worth  of  $  105,  due  in  one  year 
without  interest,  when  money  is  worth  5%  interest? 

5.  When  money  can  be  loaned  at  7%,  which  is  worth  the 
more,  $  100  at  the  present  time,  or  a  note  of  $  107  without 
interest,  due  in  one  year  ? 


268  INTEREST. 

6.  What  sum  should  be  deducted  from  a  debt  of  $  108, 
due  without  interest  in  one  year,  in  consideration  of  its 
being  paid  now,  when  money  can  be  loaned  at  8  %  ? 

366.  True  Discount  is  a  deduction  of  interest  for  the  pay- 
ment of  a  debt  before  due. 

367.  The  Present  Worth  of  a  debt  due  at  a  future  time  is 
a  sum  which  will  amount  to  the  debt  if  put  at  interest  till 
that  time. 

The  debt  is  therefore  the  amount  of  the  present  worth  for 
the  given  time. 

368.  The  true  discount  is  the  difference  between  the  debt 
and  its  present  worth.  It  is  the  interest  of  the  present 
worth  for  the  given  time. 

7.  What  is  the  present  worth  and  the  true  discount  of 
a  debt  of  $  582.40,  due  in  8  months  without  interest,  when 
money  is  worth  6%? 

Solution.  —  $  582.40  4-  f  1.04  =  $  560,  present  worth. 

$582.40  -  $560  =  $22.40,  true  discount. 
Since  $1.04  is  the  amount  of  $1  for  8  mo.,  $582.40  is  the  amount 
of  as  many  dollars  as  $  1.04  is  contained  times  in  $  582,40  =  $  560. 

Rule.  —  To  find  the  present  worth,  divide  the  debt  by  the 
amount  of  $  1  for  the  given  time. 
To  find  the  true  discount,  subtract  the  present  worth  from 
the  debt. 

8.  What  is  the  present  worth  and  true  discount  of  $400, 
due  in  one  year,  when  money  is  worth  5%  ? 

9.  A  father  wills  his  two  sons  $3000  each,  to  be  paid  in 
three  years  from  the  time  of  his  death.  What  is  the  value 
of  the  legacies  at  the  probate  of  the  will,  if  money  is  worth 
6%? 

10.  What  is  the  present  worth  of  $450,  due  in  two  years 
at  5%  ? 


BANK  DISCOUNT.  -269 

11.  What  is  the  present  worth  of  $250.51,  payable  in 
8  months,  money  being  worth  6%  ? 

12.  Which  is  better,  to  buy  flour  for  $  5  cash,  or  for 
»f  5.25  on  6  months'  time,  when  money  can  be  borrowed  at 

13.  Find  the  present  worth  of  $  750  for  6  months,  money 
being* worth  6%. 

14.  What  is  the  present  worth  of  $  600,  due  in  1  year 
without  interest,  money  being  worth  6%  ? 

15.  Write  the  note  which  would  be  given  for  the  above 
debt. 

16.  A  man  wishing  to  buy  a  house  and  lot  has  his  choice 
between  paying  $  5400  in  cash,  or  $  4000  in  cash  and  $  1700 
in  two  years.  With  money  at  6%,  which  is  the  most  ad- 
vantageous for  him  ? 

17.  Which  would  be  more  profitable,  and  how  much,  to 
pay  $4000  cash  for  a  house,  or  $4374.93  in  3  yr.  6  mo., 
money  being  worth  7  %  ? 

18.  I  can  sell  my  house  for  $2800  cash,  or  $3000  and 
wait  6  months  without  interest.  I  choose  the  latter.  Do  I 
gain  or  lose,  and  how  much,  money  being  worth  6%  ? 

19.  What  is  the  present  worth  of  a  debt  of  $385.31,  due 
in  5  months  15  days,  at  6%  ? 


BANK  DISCOUNT. 

369.  When  the  holder  of  a  negotiable  note  wishes  the 
money  before  it  becomes  due,  he  may  take  it  to  a  commercial 
bank ;  and  if  the  banker  is  satisfied  that  the  parties  to  the 
note  are  responsible,  he  will  pay  the  holder  the  amount  due 
after  deducting  the  discount.  By  this  act  the  bank  becomes 
the  holder  of  the  note,  and  at  its  maturity  the  maker  must 
pay  to  the  bank  instead  of  to  the  payee. 


270  INTEREST. 

370.  The  Maturity  Value  of  a  note  is  the  amount  due  at 

maturity. 

The  Bank  Discount  is  the  simple  interest  on  the  maturity 
value,  reckoned  from  the  day  of  discount  to  the  day  of 
maturity. 

371.  The  maturity  value  less  the  bank  discount  is  called 
Proceeds,  or  Avails.  The  time  from  the  day  of  discount  to 
the  day  of  maturity  is  called  the  Term  of  Discount. 

372.  The  maturity  value  of  a  note  not  bearing  interest  is 
the  face,  and  the  maturity  value  of  an  interest-bearing  note 
is  the  face  plus  the  interest. 

Note  1. — Only  short-time  notes  are  discounted  at  banks,  usually 
not  exceeding  4  months. 

Note  2.  — Banks  generally  require  that  the  paper  which  they  dis- 
count he  made  payable  at  some  bank. 

Note  3.  —  Banks  usually  reckon  discount  for  the  exact  number  of 
days  in  the  term  of  discount,  although  the  time  in  a  note  may  be 
expressed  in  months.     Banks  usually  regard  the  year  as  360  days. 

Note  4.  —  In  States  having  days  of  grace,  the  day  of  maturity  is 
the  last  day  of  grace. 

Note  5.  — If  the  day  of  maturity  falls  on  Sunday  or  a  legal  holiday, 
the  preceding  day  is  the  day  of  maturity  in  most  States.  In  some 
States,  however,  the  note  does  not  fall  due  until  the  day  following. 

Note  6.  —  When  a  note  is  discounted  at  date,  the  term  of  discount 
is  the  time  of  the  note  ( -I-  3  days  of  grace  in  States  having  days  of 
grace). 

Unless  otherwise  stated,  a  note  is  to  be  discounted  at  date. 

1. 
1555^  Buffalo,  N.Y.,  Jan.  15,  1896 

Two  months  after  date,  for  value  received,  /promise 

to  pay^, Richard  Turner, ^^.^^^^.^.^^^oy  order,  Three 

hundred  twenty-jive  and  ^  Dollars  at  the  Third  National 

Henry  P,  Warner, 


BANK   DISCOUNT.  271 

If  this  note  was  discounted  at  6%  at  a  bank  on  the  day 
it  was  made,  how  much  did  the  bank  deduct  ?  How  much 
were  the  proceeds  ? 

Solution.  —  The  term  of  discount  is  from  Jan.  15  to  Mar.  15  = 
Jan.  Feb.  Mar. 

16  da.  4-  29  da.  +  15  da.  =  60  da. 

The  bank  discount  is  the  interest  of  $325j3^*q   for  60  da.,  at 
6  %  =  $  3.25.     The  proceeds  =  $  325.24  -  $  3.25  =  1 321.99. 

2.  Copy  the  above  note,  and  properly  indorse  the  payee's 
name. 

3.  What  would  be  the  bank  discount  and  proceeds  of  the 
above  note  if  it  contained  the  words  "  with  interest "  ? 

Solution.  — 

Maturity  value  =  face  +  interest,  or  $325.24  +  -$3, 25  =  $328. 49. 
The  bank  discount  =  6  %  of  $328.49  for  60  da.  =  $3.28. 
The  proceeds  =  $328.49  -  $3.28  =  $325.21. 

4. 

^387'^  Boston,  Mass.,  June  27th,  1901 

Three  months  after  date,  /  promise  to  pay  to  the 
order  of_____^,^Jawes  Gr.  Rogers Three  hun- 
dred eighty-seven  and  ~  Dollars,  value  received,  at  the 
First  National  Bank,  with  interest  at  5%. 

G-eorge  Price. 
Discounted  July  27  at  5%. 
Day  of  maturity,  Sept.  27. 

Solution.  — 
Maturity  value  =  f ace  +  interest  for  90  da.,  at  5%  =  $392.34. 
Discount  at  5%  from  July  27  to  Sept.  27,  62  da.  =  $3.38. 
Proceeds  $392,34  -  $3.38  =  $388.96, 


272  INTEREST. 

5, 

$648.15  Buffalo,  N.Y.,  Jan.  31,  1900 

One  month  after  date,  I  promise  to  pay  to  the 
order  of ^.^^James  B.  Strong  ^.^^^^^^^^^.^^Six  hun- 
dred forty -eight  and  ~  Dollars,  at  the  Shoe  and 
Leather  Bank,  value  received,  with  interest  at  6%. 

Discounted  at  date  at  6%. 

The  above  note  is  interest-bearing,  therefore  the  discount 
must  be  computed  on  the  amount  at  maturity. 

6. 

$3000  Detroit,  Mich.,  Oct.  i,  1901 

Ninety    days    after    date,    for   value    received,    I 

promise   to  \)^y Jerome  K.  iVz;ro?i,___,__.,_or 

order.    Three  thousa7id   dollars,   at  the    First  National 

Leroy   O.  Bondy, 


7. 

$438.29  '  St.  Louis,  Mo.,  Feb.  7,  1902 

Two    months    after    date,    for    value    received,    I 

promise    to    pay_______^i^^A    2%omjoso?^,_______or 

order.  Four  hundred  thirty-eight  and  ^  Dollars,  at  the 

Chemical  National  Bank. 

M.  H.   Winthrop, 

Discounted  March  10  at  6%. 


BANK   DISCOUNT.  273 

8. 

$7.89§^  Cleveland,  O.,  Mar.  4,  1900 

Four  months  after  date,  I  promise  to  pay  to  the 

order    of  _______  TF!      W.     W oodf or d,^^.^.^.^^^^,^.^..^^^  Seven 

hundred    eighty -nine    and    ^  Dollars,    value    received, 

at  the  City  Bank.  ^^.     -r,     ,^ 

•^  Otis  It.    Young. 

Discounted  May  4  at  6%. 

9. 

$4920  Brooklyn,  N.Y.,  Apr.  5,  1902 

Ninety  days  after  date,  for  value  received,  /  promise 
to  pay  to  the  order  oi^.^.^^,.^.^^.^^.^Dewitt  Xo7i^______^ 

Four  thousand  nine  hundred  and  twenty  Dollars,  at  the 
Merchants'  Bank,  with  interest. 

.  Elizabeth  R.  Prentiss, 
Discounted  at  date  at  6%. 

10, 

$1312  Boston,  Mass.,  May  2,  1901 

Sixty  days  after  date,  for  value  received,  /  promise 

to  pay  to  the  order  of Edgar  JY.  Wilson . 

One  thousand  three  hundred  and  twelve  Dollars,  at  the 

First  National  Bank. 

Frank  L.  Barker, 
Discounted  May  10  at  4%. 


274  INTEREST. 

11. 
$2142.84  Albany,  N.Y.,  Dec.  15,  1900 

Five  months  after  date,  for  value  received,  I  promise 

to  pay Charles  R.  >S'H/i?igr,,_^.,,._^^_^or  order, 

Tivo   thousand  one   hundred  forty-two    ^-^  Dollars,  with 

interest,  at  the  Park  Bank. 

M.  H,  Dixon. 

Discounted  Feb.  27,  1901,  at  6%. 

12. 

$2000  San  Francisco,  Cal.,  June  10,  1900 

Three  months  after  date,  for  value  received,  /promise 
to  ^cxry^^^^^^^^^^^^^^^ Seymour  D.  Wileox,^^^^.^^^^^^,^^^,^^^^OT  order, 
Two  thousand  Dollars,  at  the  Citizens'  Bank. 

P.  J.  Reed. 

Discounted  July  10  at.  8%. 

13. 

$2500  Syracuse,  N.Y.,  July  6,  1901 

Two  months  after   date,  I  promise  to  pay  to  the 

order  oi.^^^^^.^^^^^Rohert  M.  Beecher, ^Tivo  thousand 

jive    hundred   Dollars,    at    the    Third  National   Bank. 

Value  received. 

John   Q.  Adams. 
Discounted  at  6%  at  date. 


BANK   DISCOUNT.  275 

14.  A  note  for  $  135  is  given  for  90  days,  and  discounted 
the  day  it  is  given  at  6%.     What  are  the  proceeds  ? 

15.  William  Johnson  gave  John  Doe  a  note  payable  to 
the  Binghamton  Trust  Co.,  time  60  days,  amount  $  204.60. 
Write  this  note.  After  20  days  Doe  put  the  note  in  the 
bank.     What  are  the  proceeds  of  the  note  ? 

In  the  following  problems,  write  the  notes  in  full,  and 
properly  indorse  them,  using  any  names  for  payer  and 
payee. 

16.  Find  the  bank  discount  of  $400  for  3  months  at  8%. 

17.  What  are  the  proceeds  of  $250,  with  interest  at  6%, 
discounted  at  bank  for  60  days  at  6%? 

18.  What  will  be  the  proceeds  of  a  note  for  $  175  drawn 
at  4  mo.,  with  interest  at  ^j%,  if  the  bank  discount  is  10% 
per  annum  ? 

19.  On  the  first  day  of  January,  1896,  a  farmer  gave  his 
note  at  90  da.  for  $525,  with  interest  at  6%.  When  did 
the  note  become  due,  and  what  were  the  proceeds  of  the 
note  if  discounted  at  a  bank  at  1%  a  month  on  the  tenth 
day  of  February  ? 

373.  To  find  the  Face  of  a  note,  when  the  Proceeds,  Time, 
and  Rate  are  known. 

1.  What  must  be  the  face  of  a  60-day  note,  without 
grace,  which  after  being  discounted  at  6%  will  give  $500 
as  proceeds  ? 

Solution.  — 

The  bank  discount  of  $1  at  6 %  for  60  da.  =  $.01. 
The  proceeds  of  $1  =  §1.00  -  $.01  =  $.99. 

Since  |.99  is  the  proceeds  of  $1,  $500  must  be  the  proceeds  of  as 
many  dollars  as  $.99  is  contained  times  in  $500  =  505.05  +.     Aiis. 


276  INTEREST. 

Therefore,  1505.05+  must  be  the  face  of  a  60-day  note  which  will 
give  f  500  as  proceeds  after  being  discounted  at  6  %. 

Rule.  — Divide  the  x>roceeds  by  the  jyroceeds  of  $1. 

2.  A  person  must  use  $  250  today.  For  how  much 
must  he  make  a  bank  note  for  three  months  that  will  give 
$  250  proceeds,  without  grace  ? 

3.  What  must  be  the  face  of  a  60-day  note,  payable  at  a 
Boston  bank,  upon  which  I  can  realize  f  350  after  it  is  dis- 
counted at  6%  ? 

4.  If  you  buy  goods  for  $  1200  cash,  how  large  a  not« 
payable  in  90  days,  at  6%  bank  discount,  must  you  make 
that  the  proceeds  shall  pay  for  the  goods  ?     Without  grace. 

6.  Find  the  face  of  a  60-day  note  that  will  yield  $800 
when  discounted  at  bank  at  7%,  with  grace. 

6.  How  large  a  note  must  I  make  at  a  bank  for  30  days 
to  pay  a  debt  of  $  475,  without  grace  ? 

7.  Wishing  to  borrow  $495  at  a  Chicago  bank,  for  what 
sum  must  I  make  my  note  at  60  da.,  with  interest  at  6%,  in 
order  to  obtain  this  amount?     Discount  at  ^%  a  month. 

8.  The  proceeds  of  a  Buffalo  note  at  60  da.,  when  dis- 
counted at  a  bank  at  6%  per  annum,  is  $  742.50.  What  ig 
the  face  ? 

REVIEW  OF  DISCOUNT. 

374.  1.  Define  Discount;  True  Discount;  Bank  Dis- 
count ;  Proceeds ;  Present  Worth. 

2.  How  is  the  present  worth  found  ?  The  true  dis- 
count ?     The  bank  discount  ?     The  proceeds  ? 

3.  What  is  the  term  of  discount,  and  how  is  it  found? 
The  day  of  discount,  and  how  found  ? 

4.  How  is  the  bank  discount  of  an  interest-bearing  note 
found? 


BANK  DISCOUNT.  277 

5.  How  do  you  find  the  face  of  a  note  when  the  pro- 
ceeds, time,  and  rate  are  given  ? 

6.  When  does  a  Rochester,  N.Y.,  note  mature  if  given 
for  1  month  from  Jan.  31  ? 

7.  State  a  point  of  difference  between  true  discount  and 
bank  discount. 

8.  What  kind  of  notes  only  can  be  discounted  at 
banks  ? 

9.  Bought  a  city  lot,  and  agreed  to  pay  $  546.94  at  the 
end  of  2  yr.  6  mo.,  without  interest.  E-eceiving  some  money 
unexpectedly  after  6  months,  I  wish  to  pay  cash.  How 
much  ought  I  to  pay,  money  being  worth  6%  ? 

10.  What  is  the  present  value  and  true  discount  of 
$  973.52,  due  in  1  yr.  7  mo.  24  da.  hence,  without  interest, 
money  being  worth  8%? 

11.  A  man  has  an  offer  of  $2000  cash  for  his  house,  or 
$2100  payable  in  8  months.  If  money  is  worth  8%,  which 
is  the  better,  and  how  much  ? 

12.  Find  the  discount  and  proceeds  of  a  note  for  $13,500 
payable  at  a  bank  in  90  days  after  date  without  grace,  dis- 
counted at  5%. 

13.  For  what  sum  must  C.  F.  Norton  draw  his  note  on  a 
Bingham  ton  bank,  that  when  it  is  discounted  at  4%  for  60 
days  he  will  have  $800? 

14.  A  man  owes  me  $  2540  due  in  2  years  3  months,  with- 
out interest.  If  he  pays  it  at  once,  what  discount  should  I 
allow  him  ? 

15.  Find  the  discount  and  proceeds  of  a  note  on  a  Brook- 
lyn bank  for  $350,  given  May  12,  1896,  for  4  months,  and 
discounted  at  6%,  July  15. 


278  INTEREST. 

16. 

%860  St.  Louis,  Mo.,  May  5,  1901 

Three   months   after   date,    for   value    received,    I 

promise  to  pay R.  B.   White, or  order, 

Mght  hundred  sixty  Dollars,  with  interest,  at  the  First 

National  Bank. 

H.  U.  Barrett. 
Discounted  June  11  at  6%. 

17.  For  what  sum  must  I  draw  a  four  months'  note  so  that 
the  proceeds  will  be  §800,  discounted  without  grace  at 
6%? 

18.  I  sell  my  horse  for  $216,  and  take  a  note  due  in  6 
months  without  interest.  If  money  is  worth  6%  per  an- 
num, what  is  the  present  value  of  my  note  ? 

19.  For  what  sum  must  I  give  my  note  for  60  days  at 
a  bank  in  order  to  receive  $650  proceeds,  money  being 
worth  8%  ? 

20.  Find  the  face  of  a  note,  discounted  for  $  2558.40  at 
8%,  for  a  term  of  72  days,  without  grace. 

STOCKS  AND  BONDS. 

375.  Many  kinds  of  business  require  so  much  capital 
that  several  persons  must  unite  to  raise  the  necessary 
amount. 

376.  The  Capital  to  be  raised  is  divided  into  Shares, 
usually  of  $100  each. 

Shares  are  then  sold  until  the  required  amount  is  raised. 
Each  purchaser  of  shares  is  a  Stockholder,  and  receives 


STOCKS   AND   BONDS.  279 

a  Certificate  of   Stock,  which  shows   the  number  of   shares 
purchased,  and  their  value. 

This  value  is  called  the  Par  or  Nominal  Value. 

377.  The  Market  Value  of  stocks  is  the  price  for  which 
they  are  sold. 

378.  The  value  of  stocks  depends  upon  the  profitable- 
ness of  the  business.  When  the  business  is  very  profitable, 
the  shares  are  worth  more  than  par ;  they  are  then  above 
par,  or  at  a  Premium. 

When  the  business  is  unprofitable,  the  shares  are  not 
worth  their  par  value.  They  are  then  below  par,  or  at  a 
Discount. 

379.  1.  The  capital  of  .a  company  is  $100,000.  Into 
how  many  shares  of  $  100  each  can  this  be  divided. 

2.  A  stockholder  owns  25  shares  of  stock.  How  many 
dollars  of  stock  has  he  ? 

3.  If  at  the  end  of  a  year  there  has  been  a  net  profit  of 
$  10,000,  what  per  cent  profit  has  been  made  ? 

$  10000  is  what  %  of  $  100000  ? 
The  profits  are  divided  among  the  stockholders,  and  are 
called  Dividends. 

Note  1.  —  Dividends  are  usually  declared  semi-annually  or 
quarterly. 

Note  2.  —  When  a  10  %  dividend  is  declared,  each  stockholder 
receives  10  %  of  the  par  value  of  his  shares. 

4.  What  will  be  A's  dividend  if  he  owns  35  shares  ? 
When  there  is  a  loss,  each  stockholder  is  required  to  pay 

his  share  of  the  loss.     This  is  called  an  Assessment. 

5.  What  would  be  A's  assessment  to  meet  a  2%  loss  ? 

A  person  who  buys  or  sells  stocks  for  others  is  called  a 
Stock-broker,  and  his  commission  is  called  Brokerage. 


280  INTEREST. 

Note  1.  — Brokerage  is  usually  ^  %  or  ^  %  of  the  par  value. 

Note  2.  —  In  all  stock  transactions,  dividends,  assessments,  broker- 
age, premium,  and  discount  are  computed  on  the  par  value. 

Note  3.  —  Shares  are  sometimes  issued  at  $200,  f  250,  $50,  $25, 
or  $  10  each,  but  unless  otherwise  stated  $  100  is  considered  the  par 
value  of  a  share. 

6.  What  is  the  market  value  of  10  shares  of  bank  stock, 
when  sold  at  par  ? 

7.  What  is  the  market  value  of  50  shares  of  railroad 
stock,  at  10%  premium? 

Solution.  —The  market  value  of  1  share  is  $  100  +  $10  =  $  110. 
The  market  value  of  50  shares  is  50  times  $  110. 

8.  What  is  the  market  value  of  18  shares  of  mining 
stock  at  15%  below  par? 

Solution.  —The  value  of  1  share  is  1 100  -  $  15  =  $ 85. 
The  value  of  18  shares  is  18  times  1 85. 

Stock  Quotations  are  the  published  prices  of  stocks. 
When  railroad  stock  is  quoted  at  108,  it  means  that  it 
sells  for  8%  above  par  in  the  stock  market. 

When  it  is  quoted  at  92,  it  is  selling  at  8%  below  par. 

9.  If  I  buy  stock  at  98  and  sell  it  at  101,  what  gain  do 
1  make  on  10  shares  ? 

Note.  —  Stock  at  98  means  $  98  for  a  $  100  share,  and  stock  at  101 
means  $  101  for  a  $  100  share. 

10.  When  stock  is  quoted  at  85,  what  is  the  value  of  a 
share  ?     What  is  the  value  of  1  dollar  of  stock  ? 

11.  What  must  I  pay  for  10  shares  of  stock  at  95,  if  I 
pay  the  broker  ^%  for  doing  the  business  ? 

Solution.  —  Cost    of    1    share  =  $ 95  +  Brokerage  $^  =  $95J    or 
$95.25. 
$  95.25  X  10  =  $  952.50.    Ans. 


STOCKS   AND  BONDS.  281 

12.  If  I  sell  10  shares  of  stock  at  110,  and  pay  the 
broker  ^%,  what  do  I  receive  ? 

Solution.  —  1  share  brings  $110-$^  =  $  109f ,  or  $  109.75.    , 
10  shares  bring  10  times  $  109.75  =  $  1097.50.     Ans. 

13.  A  man  invested  $4500  in  street  railway  stock  at 
10%  discount.     How  many  shares  did  he  purchase  ? 

14.  If  I  invest  $2100  in  bank  stock  at  105,  how  many 
shares  do  I  purchase  ? 

15.  A  capitalist  bought  80  shares  railroad  stock  at  87^, 
and  60  shares  mining  stock  at  114.     Find  the  cost. 

16.  $  18200  will  purchase  how  many  shares  of  stock 
selling  at  140  ? 

17.  A  stock  company  declared  a  dividend  of  2|-%.  What 
does  A  receive,  who  owns  1500  shares  of  $  10  each  ? 

18.  How  much  is  gained  on  50  shares  of  railroad  stock 
purchased  at  98  and  sold  at  102  ? 

19.  Bought  stock  at  a  discount  of  2%,  and  sold  it  at  a 
discount  of  3%.  Did  I  gain  or  lose,  and  how  much  on  20 
shares  ? 

BONDS. 

380.  To  meet  extraordinary  expenses,  governments. 
States,  cities,  villages,  counties,  towns,  and  incorporated 
companies  sometimes  borrow  money.  The  securities  given 
by  such  corporations  are  called  Bonds. 

Bonds  bear  a  fixed  rate  of  interest,  payable  annually, 
semi-annually,  or  quarterly.  They  are  bought  and  sold  in 
the  same  manner  as  stocks. 

Bonds  are  known  by  the  rate  of  interest  they  bear :  Vir- 
ginia 6's  are  bonds  of  the  State  of  Virginia,  bearing  6%  ; 
U.  S.  4's  of  '97  are  U.  S.  bonds  bearing  4%  interest,  and 
maturing  in  1897. 


282  INTEREST. 

381.  A  Coupon  is  an  interest  certificate  attached  to  a 
bond.  At  the  expiration  of  any  interest  period,  the  coupon 
is  ci^t  off  and  used  in  collecting  the  interest,  being  worth 
the  amount  of  interest  due  on  the  bond  for  a  specified 
period. 

382.  20.  What  will  be  the  cost,  including  brokerage  at 
i%,  of  200  shares  of  C,  B.,  and  Q.  R.R.  bought  at  67|  ? 

Solution.  —Cost  of  1  share  =  $67|  +  $  i  =  $68|.  Cost  of  200 
shares  =  200  x  $  68^. 

21.  How  much,  including  brokerage  at  |^%,  must  be  paid 
for  $  5000  of  U.  S.  4's  at  llOf  ? 

Solution. —$1  of  bonds  costs  $1.10|  +  .00^  =  $1.11.  $5000 
worth  will  cost  5000  times  $1.11. 

22.  What  must  I  pay  for  f  8275  of  stock  at  10%  dis- 
count ? 

23.  What  is  the  cost,  including  broker's  commission  of 
^%,  of  150  shares  of  railroad  stock  bought  at  89^  ? 

24.  I  buy  stocks  at  5%  discount,  and  sell  at  5%  pre- 
mium.    What  per  cent  profit  do  I  make  on  the  investment? 

25.  March  10,  1896,  Western  Union  Telegraph  stock  was 
quoted  at  84|^.  How  many  shares  could  be  bought  for 
i  1020,  brokerage  i  per  cent  ? 

Solution.  —  Cost  of  1  share,  $  84|  +  |  =  $  85.  As  many  shares  can 
be  purchased  as  $  85  is  contained  in  $  1020. 

26.  How  many  shares  of  stock  at  10%  premium  can  be 
purchased  for  $  2200  ? 

27.  I  invested  f  5100  in  N.Y.  and  N.H.  railroad  stock  at 
170.     How  many  shares  did  I  purchase  ? 

28.  If  I  invest  $42400  in  5%  bonds  at  106,  what  is  my 
yearly  income  ? 

Solution. —$42400 -7- $1.06  =  140000,  par  value.  How  much  is 
5%  of  $40000? 


STOCKS   AND   BONDS.  283 

29.  If  I  invest  $  21008  in  5%  bonds  at  104,  what  will  be 
my  annual  income  ? 

30.  What  will  be  my  yearly  income  if  I  invest  $11100 
in  5%  bonds  at  92,  brokerage  ^%  ? 

31.  A  man  invests  $9500  in  Virginia  6's  at  94|,  broker- 

^-ge  -4-  % .     What  is  his  quarterly  income  ? 

32.  What  will  be  my  annual  income  if  I  invest  $  5050  in 
4%  water  bonds  at  1%  premium  ? 

33.  What  is  my  dividend  on  80  shares  of  electric-light 
stock,  when  a  5%  dividend  has  been  declared  ? 

34.  What  sum  must  be  invested  in  Chicago  5's  at  92  to 
yield  an  income  of  $  600,  brokerage  i%  ? 

Solution.  —  $  600  -e-  .05  =  $  12000,  par  value.     How  much  is  92^% 
of  $12000? 

35.  How  much  must  I  invest  in  4%  bonds  at  8%  pre- 
mium, to  secure  an  annual  income  of  $  200  ? 

36.  How  much  must  be  invested  in  city  3J's  at  8%  dis- 
count, to  secure  an  income  of  $  350  ? 

37.  How  much  telegraph  stock  must  I  sell  at  11^%  dis- 
count, brokerage  -^^j  to  realize  $  8800  ? 

38.  I  invested  through  a  broker  $  5450  in  stock  at  108-J, 
brokerage  -J-^.     How  much  did  I  purchase  ? 

39.  I  sell  through  a  broker  enough  stock  at  4J%  premium 
to  realize  $  10,475,  brokerage  i  % .     How  much  do  I  sell  ? 

40.  What  rate  of  interest  do  I  receive  on  my  investment 
if  I  buy  7%  stock  at  112? 

Solution.  — Each  share  of  stock  costs  $  112,  and  yields  $  7  interest. 
$  7  is  what  per  cent  of  $  112  ? 

41.  Stock  yielding  7%  annually  is  bought  at  111-^.    What 
annual  rate  of  income  will  it  yield  on  the  investment  ? 


284  INTEREST. 

42.  What  rate  will  6%  bonds  pay  on  the  investment  if 
bought  at  112  ? 

43.  What  is  the  rate  on  Des  Moines  4's  at  a  premium  of 

44.  What  is  the  rate  of  income  on  6's  at  90,  no  brokerage  ? 

45.  Which  is  the  better  investment,  6's  at  par,  or  5's  at 
a  discount  of  12^%  ? 

46.  How  much  must  I  pay  for  5%  stock  to  secure  annu- 
ally 7  %  on  my  investment  ? 

Solution.  —  1  share  of  5  %  stock  yields  $  5  interest  annually  ;  this 
$  5  is  7  %  of  the  cost  of  one  share.  Therefore  the  question  is,  $  5  is  7  % 
of  what? 

47.  At  what  price  must  5%  stock  be  purchased  so  that 
it  will  yield  4%  on  the  investment? 

48.  How  much  must  I  pay  for  5's  to  make  my  invest- 
ment yield  6%? 

49.  What  must  I  pay  for  city  6's  that  my  investment 
may  yield  8%  annually? 

50.  How  much  must  I  pay  for  1  share  of  3%  stock,  that 
the  dividend  may  be  4%  of  the  purchase  price  ? 

51.  How  much  will  be  my  income  if  I  invest  $2300  in 
4%  bonds  at  115  ? 

Solution.  —  $2300  -r-  .|  1.15  =  $  2000  par  value.  How  much  is  4  % 
of  12000? 

52.  What  sum  invested  in  Tennessee  6's  at  85  will  yield 
an  annual  income  of  $  1800  ? 

53.  How  much  money  must  I  invest  in  6%  stock  at  80 
to  secure  an  annual  income  of  $  3186  ? 

54.  I  want  an  income  of  $  1500.  How  much  shall  I  in- 
vest in  5%  stocks  at  25%  premium  to  secure  that  amount? 

55.  How  much  must  a  man  invest  in  a  5%  stock  at  120 
to  yield  him  an  annual  income  of  $  2500  ? 


MISCELLANEOUS.  285 

MISCELLANEOUS. 

383.  1.  At  what  premium  should  an  8%  stock  sell  to 
yield  a  6%  income? 

2.  A  man  bought  stock  at  3|-%  discount  and  sold  it  at 
2%  premium,  paying  a  brokerage  of  ^%  in  both  cases.  His 
net  profit  was  f  680.     How  much  money  did  he  invest  ? 

3.  A  man  invested  his  money  in  6%  railroad  stocks, 
and  received  $  300  semi-annually.  What  was  the  sum  in- 
vested? . 

4.  Which  is  the  better  investment,  and  how  much,  a 
4%  stock  bought  at  85,  or  a  6%  stock  bought  at  120  ? 

5.  What  rate  on  the  investment  do  7%.  stocks  pay  when 
bought  at  a  premium  of  8  %  ? 

6.  What  sum  must  be  invested  in  U.  S.  6%  bonds  to 
yield  an  income  of  $  1000  ? 

7.  What  sum  must  be  invested  in  U.  S.  6's  at  $  92^  per 
share  to  yield  a  quarterly  dividend  of  $  300  ? 

8.  At  what  price  should  8%  bonds  be  bought  to  make 
the  income  from  the  investment  equivalent  to  that  from 
6  %  bonds  at  par  ? 

9.  Which  is  the  better  investment,  4%  bonds  at  86,  or 
6%  bonds  at  105? 

10.  How  much  must  I  pay  for  a  4%  stock  that  the  in- 
vestment may  yield  me  6%  ?  For  a  7%  stock  that  the 
investment  will  yield  5%? 

11.  If  25  shares  of  stock  paying  8%  are  sold  at  150,  and 
the  proceeds  loaned  at  6%,  will  the  income  be  increased  or 
diminished,  and  how  much  ? 

12.  Bought  bonds  at  125  and  sold  them  at  110,  thereb}' 
losing  $  600.     How  many  ^  1000  bonds  did  I  buy? 


286  INTEREST. 

13.  How  many  dollars  of  stock  can  I  buy  for  $  105,000 
if  stock  is  quoted  at  120?  How  many  shares  ?  What  per 
cent  do  I  receive  on  my  investment  if  the  stock  bears  6%  ? 

14.  What  is  the  cost  of  200  shares  of  D.,  L.,  and  W.  E.R. 
at  162|?  If  it  pays  a  quarterly  dividend  of  2%,  what  is 
the  yearly  income  from  this  investment?  What  rate  does 
it  pay  on  the  investment  ? 

15.  B  invests  $  1680  in  a  stock  selling  at  112.  What 
does  he  receive  from  a  dividend  of  4%  ? 

16.  An  estate  derives  an  annual  income  of  $3600  from 
stock  that  pays  7^%.  How  many  $25  shares  does  the 
estate  own? 

AVERAGE   OF  PAYMENTS. 

384.  1.  The  use  of  $  5  for  2  mo.  equals  the  use  of  $1  for 
how  many  months  ? 

2.  The  use  of  $  10  for  6  mo.  will  balance  the  use  of  $  5 
for  how  many  months  ? 

Solution.  — The  use  of  $  10  for  6  mo.  =  the  use  of  $  1  for  60  mo. 

The  use  of  $  1  for  60  mo.  =  the  use  of  $  5  for  ^  of  60 
mo.  —  12  mo. 

3.  How  long  may  $  20  be  kept  to  balance  the  use  of  $5 
for  20  mo.  ?     $  50  for  10  mo.? 

4.  A  credit  of  $  10  for  8  mo.  equals  a  credit  of  $  20 
for  how  many  months  ? 

5.  The  interest  of  $500  for  1  year  equals  the  interest 
of  $  100  for  how  long  ?     Prove  this. 

6.  I  pay  a  debt  of  $  20  four  months  before  it  is  due. 
How  long  after  it  is  due  should  my  creditor  allow  a  debt  of 
$  40  to  remain  unpaid  ? 

A  person  owing  two  debts  due  at  different  times  may 
pay  both  at  an  intermediate  time  without  loss  to  himself 


AVERAGE   OF   PAYMENTS.  287 

or  his  creditor,  by  paying  one  of  them  before  it  is  due  and 
the  other  an  equivalent  time  after  it  is  due. 

385.  The  process  of  finding  the  time  when  several  debts 
due  at  different  times  can  be  equitably  discharged  at  one 
payment  is  called  Average  of  Payments. 

386.  The  date  of  such  payment  is  called  the  Average 
Time,  and  the  time  to  elapse  before  the  payment  is  made  is 
called  the  Average  Term  of  Credit. 

Note. — The  time  to  elapse  before  any  debt  becomes  due  is  called 
a  Term  of  Credit. 

387.  When  the  terms  of  credit  begin  at  the  same  date. 

1.    On  Jan.  8,  A  bought  goods  on  the  following  condi- 
tions : 

$  300  due  in  2  months. 

$  200  due  in  4  months. 
^  100  due  in  6  months. 

How  long  after  Jan.  8  may  the  debt  be  equitably  dis- 
charged at  one  payment  ? 

Solution. — 

A  credit  of  $  800  for  2  mo.  =  a  credit  of  $  1  for    600  mo. 

A  credit  of  $  200  for  4  mo.  =  a  credit  of  $  1  for    800  mo. 

A  credit  of  $  100  for  6  mo.  =  a  credit  of  $  1  for    600  mo. 

A  is  entitled  to  a  credit  of  $  1  for  2000  mo. 

A  credit  of  $  1  for  2000  mo,  =  a  credit  of  $  600  for  ^^  of  2000  mo., 
or  3|  mo.  =  3  mo.  10  d.,  average  term  of  credit. 

Jan.  8  +  3  mo.  10  d.  =  April  18,  equated  time.     Ans. 

Short  method. 

2  mo.  X  300  =    600  mo. 

4  mo.  X  200  =    800  mo. 

6  mo.  X  100  =    600  mo. 

6^0    )2000  mo. 

3^  mo.  =  3  mo.  10  da. 
Jan.  8  +  3  m.  10  da.  =  April  18. 


288  INTEREST.  ^ 

Note.  —  One-half  a  day  or  more  is  called  another  day.  Less  than 
^  day,  not  counted. 

Call  50^  or  more  $  1.00,     Less  than  50  ^,  not  counted. 

Rule.  —  Multiply  each  debt  by  its  term  of  credit.  The  sum 
of  the  products  divided  by  the  sum  of  the  debts  will  be  the 
average  term  of  credit. 

2.  Gates  Thalheimer  sold  a  bill  of  goods  on  the  following 
terms:  $325  due  in  60  days,  $  175  due  in  90  days,  and 
$  185  due  in  4  months.     What  is  the  average  term  of  credit  ? 

3.  A  merchant  bought  $  1000  worth  of  goods,  and  agreed* 
to  pay  for  them  as  follows  :  f  100  cash ;  $  300  in  3  mo. ; 
$  250  in  4  mo. ;  and  the  balance  in  5  mo.  In  what  time 
could  he  equitably  pay  the  entire  amount  ? 

4.  On  the  first  day  of  April,  1895,  a  man  gave  3  notes, 
one  for  $  250  due  in  30  da.,  one  for  $  375  due  in  40  da.,  and 
one  for  $  425  due  in  60  da.  What  is  the  average  term  of 
credit,  and  when  could  they  have  all  been  paid  at  once  ? 

5.  D.  McCarthy  &  Co.  sold  goods  amounting  to  $4000, 
payable  as  follows :  ^  in  3  months,  J  in  4  months,  and 
the  balance  in  5  months.  What  was  the  average  term  of 
credit  ? 

6.  A  merchant  sold  goods  on  the  following  terms  :  \  pay- 
able in  2^-  months,  \  in  3i  months,  \  in  5J  months,  and  the 
balance  in  6  months.     What  was  the  average  term  of  credit  ? 

7.  Equate  the  following  payments :  $  580.75  due  in  30 
days,  $  650.25  due  in  60  days,  $  450.36  due  in  90  days,  and 
$  600  due  in  5  months. 

8.  On  the  1st  of  May  a  merchant  bought  goods  amount- 
ing to  f  1500,  agreeing  to  pay  for  them  as  follows  :  $  521.35 
on  the  10th  of  June,  $398.84  on  the  16th  of  July,  $  199.60 
on  the  15th  of  August,  and  the  balance  on  the  1st  of  Sep- 
tember.    Upon  what  date  can  he  pay  the  whole  amount  ? 


AVERAGE   OF   PAYMENTS.  289 

9.  Jacob  Amos  sold  a  bill  of  flour  amounting  to  $  2500, 
payable  as  follows :  $  500  due  in  4  months,  $  600  due  in  5 
months,  and  the  balance  due  in  6  months.  What  was  the 
equated  time  ? 

10.  A  purchased  a  farm  for  $  3000,  agreeing  to  pay  for 
it  as  follows :  $  500  cash,  $  600  in  5  months,  $  1000  in  8 
months,  and  $  900  in  1  year.  He  decides  to  give  a  note  for 
the  whole  amount.     When  was  the  balance  to  be  paid  ? 

388.    When  the  terms  of  credit  begin  at  different  dates. 

1.    A  purchased  goods  of  Dey  Bros.  &  Co.,  as  follows : 

Jan.      8,  1895.     $  200-on  2  months'  credit. 

Feb.    16,  1895.     $  400  on  3  months'  credit. 

April    4,  1895.     $  300  on  4  months'  credit. 

Find  the  average  time. 

Note.  — First  find  the  date  when  each  item  is  due. 
$200  due  Mar,   8.     200 


400  due  May  16. 

400  X 

69  da. 

=  27600 

300  due  Aug.   4. 

300  X 
900 

149  da. 

=  44700 
72300 

72300  -  900  =:  80^ 

:da.  = 

80  da. 

March  8  +  80  da. 

=  June  27,  average  time 

bt  is  due  March  8 

,  and  the  last  Aug.  4. 

The 

average 

time,  therefore,  will  be  between  these  dates. 

$  200  due  March  8  has  no  longer  time  to  run. 
$400  due  May  16  has  69  days  after  March  8. 
$  300  due  Aug.  4  has  149  days  after  March  8. 

A  is  therefore  entitled  to  a  credit  of  $  1  for  72300  da.  after  March  8, 
which  is  equal  to  a  credit  of  $  900  for  80  da.  after  March  8. 

Rule.  —  Find  the  date  on  which  each  debt  becomes  due,  and 
"using  the  earliest  of  these  as  a  standard  date,  reckon  the 
time  to  each  of  the  others. 
Multiply  each  debt  by  its  time,  and  divide  the  sum  of  the 
products  by  the  sum  of  the  debts. 


290  INTEREST. 

The  quotient  will  he  the  average  term  of  credit,  which  add  to 
the  standard  date  to  find  the  average  time. 

2.  Four  notes  are  due  as  follows  :  March  4,  $  165;  April 
15,  $325.50;  May  9,  f  94;  June  6,  $465.  What  is  the 
average  date  of  payment  ? 

3.  A  retail  dealer  bought  the  following  bills  of  goods 
on  4  months'  credit :  April  4,  $  480 ;  April  26,  $  185.65 ; 
June  1,  $  480.16;  July  6,  $  196.  What  is  the  average  time 
for  payment  ? 

4.  Bought  goods  as  follows :  Jan.  1,  $  250  at  3  mo. ; 
Feb.  1,  I  500  at  4  mo. ;  March  11,  $  106  at  60  da.  What  is 
the  average  date  of  payment  ? 

5.  Mr.  B  owes  $  1000,  due  in  5  months;  in  2  months  he 
pays  $  600.  How  long  after  the  expiration  of  the  5  months 
should  the  remainder  be  paid  ? 

Solution.  —  $  600  has  been  paid  3  months  before  due,  which  equals 
a  credit  of  f  1  for  1800  months.  He  is  entitled  to  a  like  credit  for 
$  400  after  it  is  due.     j^^y  of  1800  mo.  =  4|  months.     Ans. 

6.  A  lady  purchased  a  piano  for  $  500  on  6  months' 
credit.  If  she  pays  $  200  cash,  how  long  after  the  expira- 
tion of  the  6  months  should  the  balance  be  allowed  to  run  ? 

7.  May  1,  1896,  a  man  buys  a  store  and  fixtures  for 
$  2650,  giving  his  note  payable  in  6  months  without  interest. 
June  15,  he  pays  $  500 ;  Aug.  1,  $  750.  When  should  the 
balance  be  paid  ? 

8.  G.  L.  Hoyt  purchased  goods  of  Mann  &  Hunter  to  the 
amount  of  $3000:  $1200  to  be  paid  June  2,  1896;  $600 
to  be  paid  July  5,  1896 ;  $  200  to  be  paid  Aug.  15,  1896. 
The  balance  will  become  due  Aug.  30,  1896.  At  what  date 
must  a  note  payable  in  3  m.  be  drawn  that  it  may  become 
due  at  the  average  date  ? 


AVERAGE   OF  PAYMENTS.  291 

QUESTIONS. 

389.     1.    Define  discount;  present  wortli;  true  discount. 
Tell  how  to  find  present  worth  and  true  discount. 

2.  Define   bank   discount;  proceeds;    day   of   maturity; 
term  of  discount. 

Tell  how  to  find  bank  discount  and  proceeds. 
Tell  how  to  find  face  of  note  when  proceeds,  time,  and 
rate  are  given. 

3.  What  is  a  stock  company  ?    What  are  stocks  ?    Bonds  ? 
Shares  ? 

4.  Define  par  value ;  market  value. 

5.  What  is  a  stock  certificate  ? 

6.  Define  dividend;  assessment. 

7.  Upon  what  are  premium,  brokerage,  dividends,  and 
assessments  reckoned  ? 

8.  What  is  the  average  of  payments  ?     Equated  time  ? 
Average  term  of  credit  ? 


KATIO    AND   PROPOETION. 


390.  Oral. 

1.  5  bears  what  relation  to  10  ?     Aris.  5  is  ^  of  10. 

2.  10  bears  what  relation  to  5  ?     Ans.  10      2  times  5. 

3.  What  part  of  16  is  4  ? 

4.  How  does  $  7  compare  with  $  14  ? 

5.  John  has  20^  and  Mary  5^.  What  is  the  relation 
of  John's  money  to  Mary's  ?     Of  Mary's  money  to  John's  ? 

6.  What  is  the  relation  of  15  to  3  ?  Of  f  8  to  $16? 
Of  28  men  to  7  men  ?     Of  2  bushels  to  2  pecks  ? 

391.  Ratio  is  the  relation  between  two  like  numbers.  It 
is  found  by  dividing  one  by  the  other ;  thus : 

The  ratio  of  4  to  8  is  4  --  8  =  f 

The  sign  of  ratio  is  ( :  ).  It  is  the  division  sign  with  the 
line  omitted: 

The  ratio  of  6  to  3  is  expressed  thus,  6:3.  It  may  also 
be  expressed  fractionally,  thus,  J. 

392.  The  Terms  of  a  ratio  are  the  two  numbers  com- 
pared. 

The  first  term  of  a  ratio  is  the  Antecedent,  and  the 
second  the  Consequent. 

In  the  ratio  6  :  12,  6  is  the  antecedent,  and  12  the  conse- 
quent. 

292 


\ 

EATio.  293 

393.  A  ratio  formed  by  dividing  the  consequent  by  the 
antecedent  is  an  Inverse  ratio. 

12  -T-  6  is  the  inverse  ratio  of  6  :  12. 

394.  The  two  terms  of  a  ratio  taken  together  form  a 
Couplet. 

395.  Two  or  more  couplets  taken  together  form  a  Com- 
pound ratio. 

A  compound  ratio  may  be  changed  to  a  sim- 
■  pie  ratio  by  taking  the  product  of  the  antece- 

5  :  5  r  =  9d  :  loO     dents  for  a  new  antecedent,  and  the  product  of 
4  :  5  J  the  consequents  for  a  new  consequent. 

Antecedent  -=-  Consequent  =  Ratio. 
Therefore,  Antecedent  -~  Ratio  =  Consequent ; 
and,  Ratio  x  Consequent  =  Antecedent. 

Multiplying  or  dividing  both  terms  of  a  ratio  by  the  same  number 
does  not  change  the  ratio. 

The  ratio  12:6  =  2. 

The  ratio  3  x  12  :  3  x  6  =  2. 

The  ratio  12  --  3  :  6  -^-  3  =  2. 

Find  the  ratio  of: 
7.-56:7  11.    3bu.  :3pk. 

8.  20:300  12.    1:4 

9.  $55:f330  13.    12  :  ^ 

10.   What  is  the  ratio  ^-^j  to  ^^  ? 

Note.  —  Fractions  with  a  common  denominator  have  the  same 
ratio  as  their  numerators.  Prove  this  in  Ex.  10,  by  multiplying  both 
terms  by  10. 

17.  TV:if  =  ?     n--^j  =  '^     if:ff  =  ? 

18.  f:|=?      3:5  =  9      4:3  =  9      2:5  =  9 

19.  Find  the  inverse  ratio  of  75  to  25.     Of  15  to  225. 

20.  16  :  (?)  =  1      14  :  (?)  =  2. 

21.  (?):5  =  4.      (?):8  =  l. 

22.  Find  the  value  of  the  compound  ratio,  p. '  ^    [•  • 


14. 

H 

:16 

15. 

i- 

1 

16. 

H 

:5f 

294  PROPORTION. 

PROPORTION. 

396.  Oral. 

23.  Give  three  fractions  having  the  same  value  as  |. 

24.  Give  two  numbers  that  have  the  same  ratio  as  5  to  10. 

25.  Give  a  fraction  equal  to  |. 

26.  Give  a  ratio  equal  to  3  :  4. 

27.  How  does  the  ratio  of  5  men  to  10  men  compare 
with  the  ratio  of  $5  to  ^10? 

28.  How  does  the  ratio  of  8  lb.  to  4  lb.  compare  with 
the  ratio  of  40^  to  20^? 

29.  Name  two  numbers  that  have  the  same  relation  as 
5  to  10.     As  4  to  24.     As  8  to  16.     As  ^  to  \. 

30.  What  number  has  the  same  relation  to  5  as  12  to  3  ? 

31.  Find  a  number  whose  ratio  to  4  equals  3  :  6. 

32.  Give  three  ratios  equal  to  $100  :  foO. 

33.  Give  any  two  ratios  that  equal  each  other,  and 
express  their  equality. 

397.  An  equality  of  ratios  is  a  Proportion.  Thus, 
4 :  2  =  12  :  6.  The  ratio  of  4  to  2  equals  the  ratio  of  12 
to  6. 

A  proportion  is  usually  expressed 'with  the  sign  (::) 
between  the  ratios ;  thus,  4  :  2  : :  12  :  6.  This  is  read  4  is 
to  2  as  12  is  to  6. 

A  proportion  has  four  terms,  of  which  two  are  antece- 
dents and  two  are  consequents.  Each  term  is  a  propor- 
tional. 

398.  The  first  and  fourth  terms  are  called  Extremes,  and 
the  second  and  third  terms  are  called  Means. 

Note.  —  In  the  proportion  2  : 6  :  :  6  :  18,  the  two  means  are  the 
same  number,  6.     The  6  is  called  a  mean  proportional. 


SIMPLE  PROPORTION.  295 

Principle. — The   product  of   the  extremes  equals  the 
product  of  the  means. 

Rule.  —  To  find  an  extreme,  divide  the  jn^oduct  of  the  means 
by  the  given  extreme. 
To  find  a  mean,  divide  the  product  of  the  extremes  by  the 
given  mean. 

Supply  the  missing  term : 

34.  1:836::25:(     ).    39.    10  yd.  :  50  yd. : :  f  20  :  (|  ). 

35.  6  :  24  : :  (     )  :  40.      40.    15  lb.  :  60  lb.  : :  ($  )  :  $  12. 


36.  (     ):15 

37.  25 :(     ) 

38.  6:4::i 


:  60  :  6.     41.    i  da.  :  (  da.)  : :  12  :  6. 

:4:8.       42.    (    men)  :  75  men : :  ^  50 :  $  150. 

(     ).  43.    $f:$3f::(     )  :  5. 


SIMPLE   PROPORTION. 

399.  An  equality  of  two  simple  ratios  is  a  Simple  Pro- 
portion. 

It  is  employed  in  solving  questions  having  three  given 
terms,  two  of  which  have  the  same  relation  to  each  other 
as  the  third  to  the  required  term. 

44.    If  12  bushels  of  oats  cost  f  4,  what  will  60  bushels  cost  ? 

Solution. — There  must  be  the  same  relation  between  the  cost 
of  12  bu.  and  the  cost  of  60  bu.  as  exists  between  12  bu.  and  60  bu. 
We  place   $4   for  the   third   term.     The 
12  :  60  : :  $  4  :  ($     )      answer  will  be  the  fourth.     We  must  now 
ac\^A  form  a  ratio  of   12  and  60  that  shall  equal 

— — —  =  $20.  the  ratio  of  f4  to  the  answer.     Since  the 

third  term  is  less  than  the  required  answer, 
the  first  must  be  less  than  the  second,  and  we  have  12  :  60  for  the 
first  ratio.  The  product  of  the  means  divided  by  the  given  extreme 
will  give  the  other  extreme,  or  $20.     Ans. 

By  analysis,  —  Since  12  bu.  cost  $4, 

1  bu.  will  cost  $1^,  and 
60  bu.  will  cost  $20.    Ans. 


296  SIMPLE   PROPORTION. 

Rule.  —  Consider  the  required  answer  as  the  fourth  term^ 

and  place  the  number  that  is  like  it  for  the  third  terni. 
Place  the  two  remaining  terms  as  folloivs  : 
If  the   answer  is  to   be   larger  than   the   third   term,   the 

second   must   be   larger  than  the  first.      If  smaller,  the 

second  must  be  smaller  than  the  first. 
Divide  the  product  of  the   means   by  the  given  extreme. 

Cancel  when  possible. 

45.  If  10  sheep  cost  $35,  what  will  23  sheep  cost? 
What  will  6  sheep  cost? 

46.  If  5  men  can  do  a  piece  of  work  in  9  days,  how  long 
will  it  take  15  men  to  do  the  same  work  ? 

Solution.  —  Place  9  days  for  the  third  term,  because  it  is  like  the 
required  answer,  thus, 

:  :  9  da.  :  (    da.) 

Since  5  men  can  do  it  iii  9  days,  15  men  can  do  it  in  less  time. 
Therefore,  since  the  answer  is  to  be  smaller  than  the  third  term, 
place  5  men  for  the  second,  and  15  men  for  the  first.  Multiplying 
and  dividing  we  have  3  days.    Ans. 

47.  If  14  horses  eat  36  tons  of  hay  in  a  certain  time,  how 
many  tons  will  13  horses  eat  in  the  same  time  ? 

48.  If  it  costs  $400  to  lay  80  rods  of  street-car  track, 
how  much  will  it  cost  to  lay  3|-  miles  at  the  same  rate  ? 

49.  If  a  pole  8  ft.  high  casts  a  shadow  41  ft.  long,  how 
high  is  a  tree  which  casts  a  shadow  48  ft.  long  ? 

50.  If  a  man  walks  280  miles  in  8  days,  how  many  days 
ought  it  to  take  him  to  walk  420  miles  ? 

51.  If  it  costs  $  13.20  to  supply  a  new  arithmetic  to  each 
of  a  class  of  24  pupils,  what  will  be  the  expense  of  furnish- 
ing one  to  each  of  a  class  of  19  ? 

52.  How  far  can  a  train  run  in  3  hours,^  if  it  can  run  160 
Km.  in  4  hours  ? 


WBITTEN  EXERCISES.  297 

53.  How  many  men  will  be  required  to  do  in  10  days 
what  15  men  can  do  in  30  days  ? 

54.  What  will  8  tons  of  coal  cost,  when  17|-  tons  cost 

$  78.75  ? 

55.  If  a  certain  sum  of  money  yields  $  360  interest  in 
one  year,  what  would  the  interest  of  the  same  sum  be  for 
15  months  ? 

56.  If  $  800  yield  $  48  interest  in  a  certain  time,  how 
large  a  sum  will  yield  $  216  in  the  same  time  ? 

57.  If  the  interest  of  f  3600  for  a  certain  time  is  $  216, 
what  will  be  the  interest  of  $  800  for  the  same  time  ? 

58.  If  a  garrison  of  _240_  soldiers  have  a  supply  of  food 
sufficient  for  150  days^  how  .leng  would  the  same  food  last 
if  the  garrison  were  increased  to  6p0  men  ? 

59.  In  the  above  example,  how  long  would  the  food  last 
if,  80  men  were  sent  away  ? 

60.  Find  the  cost  of  1^-f-  bushels  of  wheat,  if  -f-  bu.  cost 

61.  If  120  shoemakers  make  40  dozen  pair  of  shoes  in 
a  certain  time,  how  many  shoemakers  would  it  require  to 
make  the  same  number  of  shoes  in  one-half  of  the  time  ? 

62.  If  a  train  runs  140  miles  in  4  hr.  30  min.,  what  is  the 
rate  per  hour  ? 

63.  When  5  tons  1250  lb.  of  coal  cost  $  24.75,  what  will 
be  the  cost  of  18  tons  500  lb.  ? 

64.  If  a  16-foot  board  9  inches  wide  contains  12  sq.  ft., 
how  wide  must  a  board  of  the  same  length  be  to  contain 
20  sq.  ft.  ? 

65.  It  takes  26  yards  of  carpet  1  yard  wide  to  cover  a 
floor.  How  many  yards  will  it  take  if  the  carpet  is  but  27 
inches  wide  ? 


298  COMPOUND  PROPORTION. 

66.  The  ratio  of  Simon's  pay  to  Matthew's  is  |.  Simon 
earns  f  18  per  week.     What  does  Matthew  earn  ? 

67.  25  men  can  do  a  piece  of  work  in  70  days ;  but  after 
30  days,  15  of  them  refuse  to  work.  In  how  many  days 
can  the  rest  complete  the  work  ? 

COMPOUND  PROPORTION. 

400.  An  equality  between  a  compound  and  a  simple  ratio 
is  a  Compound  Proportion ;  thus, 

8:4    )  . 

.     >- : :  12 :  20  is  a  compound  proportion. 

Find  the  fourth  term. 

Solution.  —  First  changing  to  a  simple  propor- 
3:6)  tion,  we  have, 

4.8j''^-(      )  3x4:6x8::3:(     ). 

Then  divide  the  product  of  the  means  by  the  given 
extreme,  using  cancellation.     Thus, 

^^^x^^l2.     Ans. 

1.  If  5  men  earn  $72  in  8  days,  how  much  can  10  men 
earn  in  6  days  ? 

Solution.  —  Since  the  answer  is  to  be  in  dollars,  place  $  72  for 

the  third  term,  and  arrange  the 
5  men  :  10  men  )  . .  a?  70  .  /  n  *®^°^^  °^  ^^^^  couplet  according 
8  days  :  6  days    f  •  *  ^  '       ^     y     as  the  answer  should  be  greater 

or  less  than  the  third  term  if  it 
depended  on  that  couplet  alone. 
Since  5  men  earn  ^  72,  10  men  can  earn  more,  so  we  place  10  men 
for  the  second  and  5  men  for  the  first ;  and  since  they  earn  $  72  in  8 
days,  they  will  earn  less  in  6  days,  so  we  place  6  days  for  the  second 
term,  and  8  days  for  the  first.  Dividing  the  prodilct  of  the  means  by 
the  extremes^  we  have, 

9       2 
iZ^.xJix_6^$108.    Ans. 


WRITTEN  EXERCISES.  299 

By  analysis. 

Since  5  men  in  8  days  earn  $  72, 

1  man  in  8  days  will  earn  $  ^^. 
1  man  in  1  day  will  earn  $  f . 
10  men  in  1  day  will  earn  $  ^-^-. 
10  men  in  6  days  will  earn  ($  108). 

Rule. —  Consider  the  answer  as  the  fourth  term,  and  place  the 

number  that  is  like  it  for  the  third. 
Arrange  the  couplets  as  if  the  answer  depended  on  each 

couplet  alone,  as  in  simple  proportion. 
Divide   the  product   of  the   means   by   the  product  of  the 

extremes.     Cancel  when  possible. 

2.  If  four  horses  eat  10  bushels  of  oats  in  5  days,  how 
many  bushels  will  be  required  to  feed  5  horses  for  2  days  ? 

3.  If  10  men  working  8  hours  a  day  can  do  a  piece  of 
work  in  12  days,  how  many  days  would  it  take  6  men,  work- 
ing 10  hours  a  day,  to  do  the  same  amount  of  work  ? 

4.  If  a  wheelman  rides  144  miles  in  3  days  of  6  hours 
each,  how  many  miles  can  he  ride  in  5  days  of  9  hours  each  ? 

5.  A  section  of  a  street  33  feet  long  and  20  feet  wide  can  r7^^   *•"" 
be  paved  with  15840  stones,  each  9  inches  long  and  8  inches         ^  (  ^ 
wide.     How  many  stones  12  inches  long  and  10  inches  wide  ^  #9^  . 
will  it  take  to  pave  a  street  12  rods  long  and  16  feet  wide  ?  X- 

6.  If  it  costs  $84  to  carpet  a  room  24  feet  long  and  21  *^-^^|<f  i 
feet  wide  with  carpet  1  yard  wide,  how  much  will  it  cost  to  ' 
carpet  a  room  25  feet  long  and  18  feet  wide  with  carpet  27 

inches  wide  ? 

7.  If  18  men  chop  360  cords  of  wood  in  12  days  of  9 
hours  each,  how  many  cords  could  17  men  chop  in  13  days 
of  10  hours  each? 

8.  If  50  men,  working  10  hours  a  day  for  11  days,  can 
dig  25  rods  of  a  canal  60  ft.  wide,  5  ft.  deep,  how  many  rods 
of  a  canal  90  ft.  wide,  7  ft.  deep,  can  140  men  dig  in  22  days 
of  8  hours  each  ? 


300  COMPOUND   PROPORTION. 

9.  If  60  men  can  build  a  wall  150  ft.  long,  64  ft.  high,  2 
ft.  thick,  in  8  days  of  10  hours  each,  how  many  days  of  8 
hours  each  will  36  men  require  to  build  a  wall  180  ft.  long, 
80  ft.  high,  2i  ft.  thick  ? 

10.  How  many  men  will  it  require  to  mow  48  acres  in  3 
days  of  12  hours  each,  if  6  men  mow  24  acres  in  4  days  of  9 
hours  ? 

11.  If  4  lb.  6  oz.  of  tea  cost  $2j\,  what  will  3  lb.  11  oz. 
cost  at  same  rate  ? 

12.  If  sufficient  flour  to  fill  8  bags  containing  98  lb.  each 
can  be  produced  from  16  bushels  of  wheat,  how  many  bushels 
will  be  needed  to  fill  14  barrels  of  196  lb.  each  ? 

13.  My  gas  bill  for  the  month  of  November  is  $3.50 
when  I  use  6  burners  3J  hours  each  evening.  How  much 
ought  it  be  for  the  month  of  December,  when  I  use  4  burners 
for  5  hours  each  evening  ? 

14.  How  long  a  piece  of  cloth  .4  m.  wide,  can  be  made 
from  175  Kg.  of  wool,  if  45  Kg.  make  a  piece  25  m.  long 
and  .6  m.  wide  ? 

15.  How  many  hours  daily  ought  30  men  to  labor  to 
perform  in  10  days  a  piece  of  work  which  is  f  as  great  as  a 
similar  job  which  25  men,  working  12  hr.  per  day,  accom- 
plished in  12  days  ? 

16.  If  $  475  yield  $  171  interest  in  6  years,  how  long  will 
it  take  $  960  to  double  itself  at  the  same  rate  ? 

17.  A  bin  8  ft.  long,  6  ft.  wide,  and  4i  ft.  deep  will  con- 
tain 270  bushels  of  wheat.  How  deep  must  another  bin  be 
built,  that  is  12  ft.  long  and  9  ft.  wide,  to  hold  405  bushels  ? 

18.  How.  many  days  ought  it  to  take  5  men  to  build  a 
wall  350  feet  long,  2-1-  feet  high,  and  3  ft.  thick,  if  10  men 
build  a  wall  315  ft.  long,  6  ft.  high,  and  2  ft.  thick,  in  30 
days? 


PARTNERSHIP. 


401.   Oral. 

1.  Charles  and  John  share  $28  in  the  ratio  of  2  to  5. 
How  much  has  each  ? 

Solution.  —  Charles  has  $  2  as  often  as  John  has  $  5.  Both  have 
$  7.  Charles  has  f  and  John  f  of  $  7.  Since  they  have  respectively 
f  and  f  of  a  part  of  $  28,  they  must  have  the  same  fractions  of  the 
whole.    Therefore, 

Charles  has  ^  of  $  28  =  $  8.    | 

John      has  f  of  $  28  zr:  $'20.  J  ' 

/   2.    Divide  30  into  two  such  parts  as  shall  be  to  each  other 
as  7  to  8. 

3.  A  man  and  a  boy  together  earn  $  48.     The  man  earns 

^  $  3  to  the  boy's  $  1.     What  is  each  one's  share  ? 

4.  A  horse  and  a  cow  were  bought  for  f  150.  The  horse 
cost  twice  as  much  as  the  cow.     What  was  the  cost  of  each  ? 

5.  Divide  140  into  four  parts  that  shall  be  to  each  other 
as  2,  3,  4,  and  5. 

6.  A  father  divided  $  7200  among  his  three  sons  in  pro- 
portion to  their  ages,  which  were  10,  12,  and  14  years  re- 
spectively.    What  was  the  share  of  each  ? 

7.  A  man  divided  f  3.60  among  three  boys,  giving  to  the 
first  5  cents  as  often  as  he  gave  6  cents  to  the  second  and  7 
cents  to  the  third.     How  much  did  each  boy  receive  ? 

301 


J 


302  PARTNERSHIP. 

8.  Professor  Adams  caught  520  fish  in  a  season,  consist- 
ing of  trout,  black  bass,  and  pickerel,  in  the  proportion  of  5, 
41,  and  3^.     How  many  of  each  kind  did  he  catch  ? 

402.  The  association  of  two  or  more  persons  in  business 
is  called  Partnership. 

The  persons  associated  are  Partners. 
The  association  is  called  a  Firm,  or  Company. 
All  the  money  or  property  furnished  by  the  partners  con- 
stitutes the  Capital. 

403.  To  find  each  partner's  share  of  the  Gain  or  Loss,  when 
their  capital  is  employed  for  the  same  time. 

1.  A,  B,  and  C  formed  a  partnership.  A  contributed  to 
the  capital  $  800 ;  B,  $  1000 ;  and  C,  $  1200.  At  the  end  of 
a  year  they  found  that  there  was  a  gain  of  $  1500.  What 
was  each  man's  share  of  the  gain  ? 

Solution.  — 

$  800  +  $  1000  +  $  1200  =  $  3000,  entire  capital. 
A's  gain,  ^%%%,  or  j%  of  8  1500  =  i$400. 
B's  gain,  ^§g,  or  j%  of  $  1500  =  $  500. 
C's  gain,  ^^fg,  or  j%  of  $  1500  =  ^600. 

Rule.  —  Take  for  each  man's  share  of  the  gain  or  loss  such 
a  part  of  the  whole  gain  or  loss  as  his_  capital  is  of  the 
whole  capital. 

2.  Mr.  Wilson  and  Mr.  Mead  entered  into  partnership. 
Mr.  Wilson's  capital  was  $3000,  and  Mr.  Mead's  $2000. 
They  gained  $1500.  What  was  each  partner's  share  of  the 
gain  ? 

3.  Messrs.  Jones  and  Smith  are  partners,  with  a  capital 
of   $3000  and  $5000  respectively.      After  one  year  they 


WRITTEN   EXERCISES.  303 

find  that  they  have  gained  $2000.     How  much  of  the  gain 
should  each  receive  ? 

4.  Three  men  form  a  partnership  at  the  same  time.  A 
invests  $1250;  B,  $2000;  C,  $1550.  They  gain  $1200. 
What  is  each  man's  share  of  the  gain  ? 

5.  Three  men  hired  a  coach  to  convey  them  to  their 
respective  homes.  A's  home  was  20  miles.  %way,  B's  24 
miles,  and  C's  28  miles.  They  paid  $24  lor  the  coach. 
What  ought  each  to  pay? 

6.  A  cargo  of  wheat  valued  at  $4500  was  entirely 
destroyed.  One-third  of  it  belonged  to  A,  two-fifths  to  B, 
and  the  remainder  to  C.  What  was  each  one's  share  of  the 
loss,  there  being  an  insurance  of  $  3600  ? 

7.  A  man  fails  in  business  to  the  amount  of  $15000, 
and  his  available  means  amount  to  only  $  9000.    How  much 

^  will  two  of  his  creditors  receive,  to  one  of  whom  he  owes 
"^  $3000,  and  the  other  $4500  ? 

8.  A  and  B  gain  in  business  $  2500,  of  which  A's  share 
is  $  1000,  and  B's  $  1500.  What  part  of  the  capital  does 
each  furnish,  and  what  is  the  investment  of  each  if  their 

H  joint  capital  is  $16000? 

9.  I  form  a  partnership  with  two  members  of  my  class. 
The  second  member  invests  a  certain  amount,  the  first 
invests  ^  as  much,  while  I  invest  as  much  as  the  other  two. 
What  share  of  the  profit  do  I  get  ? 

10.  Purchased  a  flour-mill  for  $  42000.  X's  share  of  the 
mill  was  y\,  Y's  J,  and  Z's  the  remainder.     At  the  end  of 

'  three  years  they  sold  the  mill  at  a  reduction  of  $  5000,  but 
the  profits  in  the  business  during  the  three  years  were 
$  20000.     What  was  each  man's  net  gain  ? 

11.  Two  persons  invest  in  trade  $  800.  They  gain  $  150. 
The  gain  and  stock  of  the  first  amount  to  $570.  What  is 
the  stock  and  the  gain  of  each  ? 


304  PARTNERSHIP. 

When  the  capital  of  the  partners  is  not  employed  for  the 
same  time. 

12.  A  and  B  formed  a  partnership.  A  furnished  $  500 
for  8  months,  and  B  $600  for  10  months.  They  gained 
$360.     What  was  each' partner's  gain  ? 

Solution.  t^M^ 00  f dti8  mo.  =  $  4000  for  1  mo. 
•*  B  $600  for  10  mo.  =     6000  for  1  mo. 


$  10000 
A's  share  =  ^  of  $  360  =  $  144. 
B's  share  =  j\  of  $  360  =  $  216.  * 

The  use  of  $  500  for  8  months  is  equivalent  to  the  use  of  $  4000  for  1 
month ;  and  the  use  of  §  600  for  10  months  is  equivalent  to  the  use 
of  16000  for  1  month.  Consider  A's  capital  to  be  $4000  and  B's 
$6000.     A's  share  of  gain  =  -j^o  5  ^'s  share  of  gain  =  j%. 

13.    A  commenced  business  with  $  10000  capital.      Four^ 
months  later  B  put  in  $  10500.     Their  profits  at  the  end  of 
a  year  were  $  5100.     What  was  each  man's  share  of  the 
gain? 

^     14.    Three  persons  loaned  a  sum  of  money  for  which  they  W^ 
received  $  1596  interest.     The  iirst  loaned  $  4000  for  12  mo., 
the  second  $  3000  for  15  mo.,  and  the  third  $  5000  for  8  mo. 
How  much  interest  did  each  receive  ? 

15.  A  and  B  were  in  partnership  for  2  years.  A  at  first 
invested  $  2000,  and  B  $  2800.  At  the  end  of  9  months  A 
took  out  $  700,  and  B  put  in  $  500.  They  lost  in  the  two 
years  $  3740.     Apportion  the  loss. 

16.  A,  C,  and  H  form  a  partnership.  A  puts  in  $  8000, 
C  $  5000,  H  $  10000.  A's  capital  remains  in  the  business 
8  mo.,  C's  9  mo.,  H's  12  mo.  The  net  gain  is  $6900. 
Find  each  man's  share  of  the  gain. 


WRITTEN   EXERCISES.  305 

17.  Two  partners  entered  business,  agreeing  to  continue 
for  18  months.  A  put  in  $  2000  at  first,  and  8  months 
later  $  1200  additional.  B  at  first  put  in  f  3000,  but  at  the 
end  of  4  months  drew  out  $  600.  On  closing  their  account 
they  found  they  had  made  $  2808.  What  was  each  man's 
share  of  the  gain  ? 

18.  On  Feb.  1  Messrs.  Scott  and  White  commenced  busi- 
ness with  $3000,  each  furnishing  $1500.  On  April  1 
White  put  in  $  1300  more.  On  May  1  they  took  Watson 
into  partnership  with  $  2500.  At  the  end  of  one  year, 
how  should  a  net  gain  of  $  2400  be  apportioned  ? 

19.  A's  capital  was  in  business  6  months,  B's  7  months, 
gnd  C's  11  months.  A's  gain  was  $  600,  B's  $  1400,  and  C's 
$990.  Their  joint  capital  was  $7800.  What  was  each 
man's  capital  ? 

20.  A  put  $  600  in  trade  for  5  mouths,  and  B  $  700  for  6 
months.     They  gained  $  228.     What  was  each  man's  share  ? 

21.  April  1,  1895,  A  goes  into  business  with  a  capital  of 
$  6000 ;  July  1,  1895,  he  takes  in  B  as  a  partner  with  a 
capital  of  $  8000 ;  and  Oct.  1, 1896,  they  have  gained  $  2900. 
Find  the  gain  of  each. 

22.  Three  men,  A,  B,  and  C,  hire  a  pasture  for  6  months 
for  $  75.  A  puts  in  10  cows  at  first,  but  at  the  end  of  1 
month  takes  away  4.  B  puts  in  8,  and  in  3  months  takes 
out  5,  but  adds  2  after  2  months  more.  C  puts  in  6  and 
in  4  months  he  puts  in  8  more.     What  should  each  pay  ? 

QUESTIONS. 

404.    1.^  Define  ratio ;  the  terms  of  a  ratio. 

2.    How  is  ratio  found 7"^  What  is  direct   ratio?     In- 
verse ratio  ?  >»  A  simple  ratio  ?  V  A  compound  ratio  ? 

3.y  Tell  how  to  find  ratio  when  antecedent  and  conse- 
quent are  given.^  To  find  consequent  when  antecedent  and 


306  PARTNERSHIP. 

ratio  are  given.     To  find  antecedent  when  consequent  and 
ratio  are  given. 

4:/  Define  proportion.*^  How  many  terms  in  a  simple 
proportion?*^  Name  them. 

5.^ Give  the  principle  of  proportion. 

6. "^  What  number  is  placed  for  the  third  term?^  The 
second  ?  ^  The  first  ?     How  is  the  fourth  term  then  found  ? 

7.'  What  is  a  compound  proportion?  What  number  is 
placed  as  the  third  term  ?  »  How  is  each  couplet  theii 
arranged  ?     How  is  the  fourth  term  found  ? 

8/  What  is  a  partnership  ?  *  A  company  ?  ♦ 

9.    Define  capital  stock  J*^  dividends. 

^  10.    Tell  how  to  find  each  partner's  share  of  the  profit  or 

loss  when  the  capital  of  each  is  invested  for  the  same  tim5. 

'   When  the  capital  of  each  is  not  invested  for  the  same  time. 


INVOLUTION. 


405.  1.  3x4x2  =  what  ? 

2.  3  X  3  X  3  =  what? 

Note,  —  In  Example  2  the  factors  are  equal ;  in  Example  1  they  are 
unequal. 

The  product  of  equal  factors  is  a  Power. 

3.  What  is  the  product  of  4  taken  3  times  as  a  factor  ? 

4.  What  is  the  product  of  6  taken  twice  as  a  factor  ? 

5.  What  is  the  product  of  |-  used  three  times  as  a  factor? 

6.  What  is  the  product  of  .6  used  twice  as  a  factor  ? 

406.  The  process  of  finding  powers  is  Involution. 

407.  A  power  is  named  according  to  the  number  of  its 
equal  factors. 

The  product  of  two  equal  factors  is  the  Second  Power,  or 
Square,  of  the  equal  factor. 

The  product  of  three  equal  factors  is  the  Third  Power,  or 
Cube,  of  the  factor. 

Note. — The  second  power  is  called  a  square  because  the  area  of 
any  square  figure  is  the  product  of  two  equal  factors,  length  and 
breadth  ;  and  the  third  power  is  called  a  cube  because  the  solidity  of 
any  cube  is  the  product  of  three  equal  factors,  length,  breadth,  and 
thickness. 

307 


13.   2.5' 

16.    (})' 

14.    1.1^ 

17.  (^y 

15.   .002' 

18.    (2^)^ 

308  INVOLUTION. 

408.  A  small  figure  at  the  right  and  above  a  number  to 
show  how  many  times  it  is  to  be  used  as  a  factor  is  called 
an  Exponent.     Thus, 

42  zz:  4  X  4  is  4  to  the  second  power,  or  the  square  of  4  ; 
2^  =  2  X  2  X  2  is  2  to  the  third  power,  or  the  cube  of  2  ; 
3*  =  3x3x3x3  is  3  to  the  fourth  power,  or  the  fourth  power  of  3. 

Eead :  8^  15',  5',  f^'^^,  |,  84»  16'. 

409.  Find  the  powers  : 

7.  53  10.    6' 

8.  2'  11.    1^ 

9.  252  12.    .01^ 

19.  Find  the  square  of  36. 

20.  Find  the  cube  of  15. 

21.  Find  the  4th  power  of  6. 

22.  Find  the  5th  power  of  3. 

23.  State  the  difference  between  a  power  and  any  other 
product. 

24.  Find  the  difference  between  the  square  and  the 
cube  of  9. 

25.  Find  the  difference  between  the  4th  and  5th  powers 
of  6. 

26.  Find  the  difference  between  the  square  and  the  cube 
of  1 

27.  What  is  the  difference  between  the  square  and  the 
cube  of  .15  ? 

28.  Which  is  greater  and  how  much,  the  square  or  the 
cube  of  ^? 


EVOLUTION. 


410.  1.  What  factor  is  used  3  times  to  produce  27  ? 

2.  What  are  the  two  equal  factors  of  64  ? 

3.  What  is  one  of  the  three  equal  factors  of  8  ? 

4.  36  is  the  square  of  what  number  ? 

5.  64  is  the  cube  of  what  number  ? 

6.  144  is  the  second  power  of  what  ? 

7.  1728  is  the  cube  of  what  ? 

411.  One  of  the  equal  factors  of  a  power  is  a  Root. 

One  of  two  equal  factors  of  a  number  is  the  Square  Root 
of  it. 

One  of  the  three  equal  factors  of  a  number  is  the  Cube 
Root  of  it. 

The  fourth  root  of  a  number  is  one  of  its  4  equal  factors. 

The  square  root  of  16  =  4.  The  cube  root  of  27  =  3.  The 
fourth  root  of  16  =  2. 

412.  The  Radical  Sign  (-y/)  placed  before  a  number  indi- 
cates that  its  root  is  to  be  found. 

The  radical  sign  alone  before  a  number  indicates  the 
square  root ;  thus,  V9  =  3  is  read,  the  square  root  of  9  =  3. 

413.  A  small  figure  placed  in  the  opening  of  the  radical 
sign  is  called  the  Index  of  the  root,  and  shows  what  root  is 
to  be  taken ;  thus,  V8  =  2  is  read,  the  cube  root  of  8  is  2. 

Read  the  following : 

V8l,  -^64,  -v/Sl,  Viil,  -^Tm,  ^/'9,  -s/SM 
309 


310  EVOLUTION   AND   INVOLUTION. 

EVOLUTION  AND  INVOLUTION. 

414.  1.  rind  the  square  of  11.  The  cube  of  6.  The 
fourth  power  of  5. 

2.  Eind  the  square  root  of  49.  The  cube  root  of  8. 
The  square  root  of  -f^. 

3.  92==?     -v/9  =  ?     8^  =  ?     -v^8  =  ? 

4.  Write  all  the  squares  from  1  to  100. 

5.  Write  all  the  cubes  from  1  to  1000. 

6.  Learn  the  second  and  third  powers  of  numbers  from 
1  to  12. 

SQUARE  ROOT. 

415.  The  square  of  a  number  is  the  product  of  that 
number  taken  twice  as  a  factor. 

Blackboard. 

12  =  1.  102  =  100.  1002  ^  i()000^ 

92  =  81.  902  ^  gjLOO.  9002  =  810000. 

From  the  above  illustration  it  is  seen  that  annexing  one 
cipher  to  a  number  annexes  two  ciphers  to  the  square  of 
that  number,  as  in  1^  =  1 ;  10^  =  100 ;  100^  =  10000. 

416.  A  square  contains  twice  as  many  figures  as  its 
root,  or  twice  as  many  less  one. 

Squares  of  even  tens. 

OraL 

1.  202=?  3^   802=?  5^   702  =  ?  rj^  5002  =  ?  9^   6002  =  ? 

2.  502  =  ?  4^   302=?  Q    2002=?  8.   9002=? 

417.  The  square  of  a  number  composed  of  tens  and 
units  may  be  found  as  follows  : 

24  =  20  +  4  =  2  tens  +  4  units. 
242=  (20  +  4)  X  (20  +  4). 


SQUARE  ROOT.  311 

20  +  4=    24 

20  4-4=    24 

(20  X  4)  +  42  =    96 

202  +  (20  X  4)  =  480 

202  +  2  X  (20  X  4)  +  42  =  576 

From  the  operation,  we  find  that, 

The  square  of  the  tens      .     .  202  =:  400 

2  times  the  tens  by  the  units,     2  x  (20  x  4)  =  160 

The  square  of  the  units    .     .  42  =    16 

400  +  160  +  16  =^6 

418.  Prin-ciple.  —  The  square  of  a  number  composed  of 
tens  and  units  is  equal  to  the  square  of  the  tens,  plus  twice 
the  product  of  the  tens  by  the  units,  plus  the  square  of  the 
units. 

Formula.  —  Tens2  +  2  x  tens  x  units  +  units^. 

Separate  the  following  into  tens  and  units,  and  find  their 
squares  :  15,  25,  74. 

419.  By  reversing  the  process  we  may  find  the  Square 
Root. 

10.    What  is  the  square  root  of  1225  ? 

Solution.  —  Separating  into  periods  of  two  figures  each,  begin- 
ning at  units,  we  have  12'25.  Since  there  are  two  periods  in  the 
power,  there  must  be  two  figures  in  the  root,  tens  and  units. 

The  greatest  square  of  even  tens  contained  in  1225  is  900,  and  its 
square  root  is  30  (3  tens). 

1225  I  30  +  5  =  35. 
Tens2,  302  ^        900 

2  x  tens  =  2  X  30  =60  [325 

2  X  tens  +  units  =  2  x  30  +  5  =  65  [325 

Subtracting  the  square  of  the  tens,  900,  the  remainder  consists  of 
2  X  (tens  X  units)  +  units. 

325,  therefore,  is  composed  of  two  factors,  units  being  one  of  them, 
and  2  x  tens  +  units  being  the  other.  But  tlie  greater  part  of  this 
factor  is  2  X  tens  (2  x  30  =  60).  By  trial  we  divide  325  by  60  to  find 
the  other  factor  (units),  which  is  5,  if  correct.  Completing  the  factor, 
we  have  2  x  tens  +  units  =  65,  which,  multiplied  by  the  other  factor, 
5,  gives  325.     Therefore  the  square  root  is  30  +  5  =  35. 


312  EVOLUTION. 

420.  Square  root  may  be  explained  by  the  aid  of  dia^ 
grams. 

The  area  of  every  square  surface  is  the  product  of  two 
equal  factors,  length  and  width. 

Finding  the  square  root  of  a  number,  therefore,  is  equiva- 
lent to  finding  the  width  of  a  square  surface,  its  area  being 
given. 

421.  The  following  formulas  illustrate  the  principles 
which  underlie  the  operations  of  square  root: 

1.  Length  x  Width  =  Area, 

2.  Area  -^  Length  =  Width. 

3.  Area  -f-  Width  =  Length. 

1.    Find  the  width  of  a  square  whose  area  is  1296  sq.  ft. 

Fig.   1. 


2  X  30  ft.  =  60  ft. 

2  X  30  ft.  +  6  ft.  =  m  ft. 


Solution.  —  The  great- 
est square  of  even  tens 
contained   in   1296  sq.  ft. 

is  900  sq.ft.     (Square  A)     \~  b | ^ L£J 

Its  width  is  30  ft.      1296  ,  Fig.  2. 

sq.  ft.  -  900  sq.  ft.  =  396 

sq.  ft.,  tlie  area  of  6,  c,  and  d,  considered  as  one  rectangle  (Fig.  2), 
whose  width  we  desire  to  find.  The  length  of  this  rectangle  is 
(2  X  30  ft. )  60  ft.  +  the  length  of  c.  But  we  cannot  know  the  length 
of  c  till  we  find  its  width.  By  trial  (Formula  2),  we  divide  the  area, 
396  sq.  ft.,  by  60  ft.,  its  approximate  length.  The  quotient,  if  cor- 
rect, is  6  ft.,  the  width  desired.  To  test  the  correctness:  Add  the 
6  ft.  to  the  trial  divisor,  and  we  have  Q6  ft. ,  the  entire  length  of  a,  6, 
and  c,  which  (Formula  1),  multiplied  by  its  width,  6  ft.,  gives  its  area, 
396  sq.  ft.    There  is  no  remainder,  and  the  work  is  correct.    Therefore, 

30  ft.,  the  width  of  J.  +  6  ft.,  the  width  of  a,  &,  and  c,  =  36  ft.,  the 
width  of  the  original  square. 


AREA. 

1296 

WIDTH. 

d 
o 

900 
896 

30  ft. 
Oft. 

396 

36  ft. 
Ans. 

2 

b 

c 

A 

900 

d 

sq.  ft. 

30  ft.  30  ft.  6  ft. 


SQUAEE  ROOT.  313 

Notes.  —  All  the  numbers  in  the  middle  column  denote  area. 
1296  sq.  ft.  =  area  of  the  original  square  ;  900  sq.  ft.  =  the  area  of 
A;  and  396  sq.  ft.,  the  area  of  6,  c,  and  d. 

The  numbers  in  the  left-hand  column  denote  length.  60  ft.  =  the 
approximate  length  (or  the  trial  divisor)  of  b,  c,  and  d ;  and  6Q  ft. 
the  exact  length,  or  the  complete  divisor. 

The  numbers  in  the  right-hand  column  denote  width.  30  ft.  =  the 
width  of  A  ;  and  6  ft.  the  width  of  6,  c,  and  d  ;  36  ft.  =  the  width  of 
the  original  square. 

In  dividing,  to  find  the  width  of  6,  c,  and  d,  since  the  divisor  is  too 
small,  care  must  be  taken  that  the  quotient  figure  be  not  too  large. 


SHORT    METHOD. 

422.    2.   Find  the  square  root  of  1306.0996. 


13'06'.09'96  C 36.14 
9 
66j    40e^ 
396 
721 )  1009 
721 


7224)  28896 
28896 

Rule.  —  Beginning  at  the  decimal  point,  separate  the  number 
into  periods  of  two  figures  each,  pointing  whole  numbers 
to  the  left  and  decimals  to  the  Hght.  Find  the  greatest 
square  in  the  left-hand  period,  ayid  ivrite  its  root  at  the 
right.  Subtract  the  square  from  the  left-hand  period,  and 
bring  down  the  next  period  for  a  dividend. 

Divide  the  dividend,  with  its  right-hand  figure  omitted,  by 
twice  the  root  already  found,  and  annex  the  quotient  to  the 
root,  and  to  the  divisor.  Multiply  this  complete  divisor 
by  the  last  root  figure,  and  bring  down  the  next  period  for 
a  dividend,  as  before. 

Proceed  in  this  manner  till  all  the  periods  are  exhausted. 


314  EVOLUTION. 

Note  1.  —  "When  0  occurs  in  the  root,  annex  0  to  the  trial  divisor, 
bring  down  the  next  period,  and  divide  as  before. 

Note  2.  —  If  there  is  a  remainder  after  all  the  periods  are  ex- 
hausted, annex  decimal  periods. 

Note  3.  —  If,  after  multiplying  by  any  root  figure,  the  product  is 
larger  than  the  dividend,  the  root  figure  is  too  large  and  must  be 
diminished.  Also  the  last  figure  in  the  complete  divisor  must  be 
diminished. 

Note  4.  —  For  every  decimal  period  in  the  power,  there  must  be  a 
decimal  figure  in  the  root. 

Note  5.  —  If  the  last  decimal  period  does  not  contain  two  figures, 
supply  the  deficiency  by  annexing  a  cipher. 

3.  Find  the  square  root  of  253009. 

Solution.  —  As    0 
25'30'09(5     occurs  in  the  root,  we  25'30'09(503  Ans. 

25  annex    0   to  the    trial  25 

10)"^             ^^^^''^''    ^^'   ^^^/^-     1003)~3009 
— ^  other  period  to  the  divi-     ^      oAAn 

dend,  and  divide  as  before.     Thus,  —  ^^25 

Note.  — To  find  the  square  root  of  a  common  fraction,  extract  the 
root  of  each  term  separately.  If  both  terms  are  not  squares,  change 
the  fraction  to  a  decimal,  and  then  extract  the  root.  The  result 
will  be  the  approximate  root.  Change  mixed  numbers  to  improper 
fractions. 

4.  What  is  the  square  root  of  y^  i  ?    ^       = Ans. 

V144      12 
Find  the  square  root  of : 

14.  .06432  23.  ^ 

15.  .005625  24.  n 

16.  .913936  25.  Ill 

17.  25.6036  26.  4^ 

18.  24.3049  27.  ^ 

19.  .612089  28.  ffff 

20.  329.7643217  29.  36.45f 

21.  1684.298431  30.  2863|J 

22.  389765268  31.  189J| 


5. 

8836 

6. 

15876 

7. 

370881 

8. 

46656 

9. 

820836 

10. 

29.0521 

11. 

9.2416 

12. 

3180.96 

13. 

.007921 

SQUARE   ROOT. 


315 


41.    13.2^ 


Find  the  square  root  to  four  decimal  places : 

32.  .15  35.   4.7  38.    3.67 

33.  .18  36.   72.5  39.    .222  42.    .009^^ 

34.  17  37.    119  40.    963  43.    .003f 

44.  What  is  the  length  of  one  side  of  a  square  field  that 
has  an  area  equal  to  a  field  75  rd.  long  and  45  rd.  wide  ? 

45.  How  wide  is  a-field  containing  7056  square  rods  ? 
Perform  the  indicated  operations. 

Note.  —  Carry  decimals  to  the  third  place. 


46.    V3.26  X  .0063. 


47.     03  X  V|4-f 
1        V9 
V9       3 


48. 


49.  Vi  X  f 

50.  Vi  X  Vi- 


51. 

V3.532  -  6.28. 

52. 

V4  +  62  +  2. 

53. 

VF  +  (|/- 

54. 

V625  + 1296. 

55.    V625+V1296. 


RIGHT-ANGLED    TRIANGLES. 

423.  A  triangle  having  one  right  angle  is  a  Right-Angled 
Triangle. 

424.  The  side  opposite  the  right  angle  is  the  Hypothe- 
nuse,  as  AB.  BO  is  the  Perpendicular,  and  AC  the  Base. 
In  the  triangle  ABC,  the 
hypothenuse  is  5  inches,  the 
perpendicular  3  inches,  and 
the  base  4  inches.  \  >^    v  \/\  y\  b 


C 


316  EVOLUTION. 

425.  It  will  be  seen  that  the  square  of  the  hypothenuse 
is  25  sq.  in.,  which  is  equal  to  the  square  of  the  base,  16 
sq.  in.,  plus  the  square  of  the  perpendicular,  9  sq.  in. 

Principle.  —  The  square  of  the  hypothenuse  equals  the 
sum  of  the  squares  of  the  two  shorter  sides.  Therefore, 
to  find  the  hypothenuse,  take  the  square  root  of  the  sum  of 
the  squares  of  the  base  and  perpendicular. 


VBase^  4-  Perpendicular^  =  Hypothenuse. 

426.  To  find  the  base  or  the  perpendicular,  take  the  square 
root  of  the  difference  between  the  squares  of  the  hypothe- 
nuse and  the  other  side. 


vHypothenuse^  —  Base^  =  Perpendicular. 
VHypothenuse'^  —  Perpendicular^^  =  Base. 

1.    The  base  of  a  right-angled  triangle  is  32  ft.,  and  the 
perpendicular  24  ft.     What  is  the  hypothenuse  ? 

Solution.  —  322  +  242  ^  iqqq.         -v/ieOO  =  40  ft.     Ans. 


Or,  \/322  +  242  ^  40  ft. 

2.    The  hypothenuse  of  a  right-angled  triangle  is  40  ft., 
and  the  base  32  ft.     What  is  the  perpendicular  ? 

402-322  =  576.  \/576  =  24ft.     Ans. 


Or,  V402  -  322  =  24  ft. 

3.  A  40-foot  ladder  placed  24  feet  from  a  house  will  just 
reach  to  the  top  of  it.     How  high  is  the  house  ? 

4.  What  is  the  length  of  a  ladder  that  will  reach  the 
top  of  a  house  40  feet  high,  when  the  foot  is  placed  30  feet 
from  the  house  ? 

5.  A  rope  150  ft.  long  fastened  to  the  top  of  a  flag-pole 
reaches  the  ground  40  feet  from  the  base.  How  high  is  the 
pole  ? 


SIMILAR   SURFACES.  317 

6.  What  is  the  hypothenuse  of  a  right-angled  triangle 
whose  perpendicular  is  36  feet,  and  whose  base  is  27  feet  ? 

7.  A  square  farm  contains  360  acres.  What  is  the  di- 
agonal distance  between  its  opposite  corners  ? 

8.  A  telegraph  pole  32  feet  high  casts  a  shadow  28  feet 
in  length.  What  is  the  distance  from  the  top  of  the  pole  to 
the  end  of  the  shadow  ? 

9.  The  base  of  a  right-angled  triangle  is  16  m.,  and  the 
perpendicular  is  12.8  m.    What  is  the  hypothenuse  ? 

10.  A  boy  rides  his  wheel  due  north  at  the  rate  of  15 
miles  an  hour,  and  another  boy  starting  from  the  same 
place,  rides  due  east  at  the  rate  of  18  miles  an  hour.  How 
far  are  they  apart  at  the  end  of  5  hours  ? 

11.  What  is  the  length  of  the  diagonal  of  a  floor  16  ft. 
long  and  12  ft.  wide  ? 

12.  A  crayon  box  is  6  in.  long,  4  in.  wide,  and  4  in.  high. 
What  is  the  diagonal  distance  across  the  bottom  ?  Between 
the  opposite  corners  ? 

13.  A  street  is  32  ft.  wide  from  curb  to  curb.  A  tele- 
graph pole  40  ft.  high  stands  upon  one  side  of  the  street, 
How  long  must  a  wire  be  to  reach  from  the  top  of  the  pole 
to  the  opposite  side  of  the  street  at  the  curb  ? 

SIMILAR    SURFACES. 

427.  Surfaces  having  the  same  form  without  regard  to 
size  are  Similar  Surfaces. 

Note. — Any  two  squares  or  any  two  circles  of  different  size  are 
Similar  Figures.  Rectangles,  triangles,  etc.,  are  similar  when  their 
corresponding  dimensions  are  proportional. 

Oral. 

1.  What  is  the  area  of  a  square  whose  side  is  2  ft.  ? 

2.  What  is  the  area  of  a  square  whose  side  is  3  ft.  ? 


318  EVOLUTION. 

3.  What  is  the  ratio  of  the  two  sides  ? 

4.  What  is  the  ratio  of  the  two  areas  ? 

5.  Are  these  ratios  equal  ?      (2  ft.  :  3  ft.)       (4  sq.  ft.  :  9 
sq.  ft.) 

Solution.  —  From  the  illustration  it  will 
be  seen  that  the  areas  are  to  each  other  as 
the  squares  of  the  sides  ;  not  as  2  to  3,  but 
as  4  to  9. 


2  ft. 

8  ft. 


Principles.  —  Similar  surfaces  are  to  each  other  as  the 
squares  of  their  corresponding  dimensions. 

Corresponding  dimensions  are  to  each  other  as  the  square 
roots  of  their  areas. 

6.  A  circle  is  4  inches  in  diameter ;  another  is  8  inches 
in  diameter.     What  is  the  ratio  of  their  areas  ? 

7.  A  circle  has  an  area  of  16  square  feet;  another  has 
an  area  of  64  square  feet.  What  is  the  ratio  of  their 
diameters  ? 

8.  The  area  of  a  rectangle  12  ft.  long  is  84  square  feet. 
What  is  the  area  of  a  similar  rectangle  6  feet  long  ? 

9.  Two  similar  fields  have  areas  of  12  acres  and  8  acres 
respectively ;  the  larger  is  32  rods  wide.  How  wide  is  the 
smaller  ? 

10.  The  altitudes  of  two  similar  triangles  are  20  ft.  and 
10  ft. ;  the  area  of  the  smaller  is  80  square  feet.  What  is 
the  area  of  the  larger  ? 

CUBE    ROOT. 

428.  The  cube  of  a  number  is  the  product  of  that  number 
taken  three  times  as  a  factor. 

Blackboard. 

V  =  1.  10^  =  1000.  1003  ^  1000000. 

9^  =  729.  908  =  729000.  9003  =  729000000.    . 


CUBE  ROOT.  319 

429.  Annexing  one  cipher  to  a  number,  annexes  three 
ciphers  to  the  cube  of  the  number,  as  shown  in  1^,  10^, 
1003,  etc. 

430.  Cubes  of  even  tens. 

1.  103  =  ?  4.     403=?  7.    3003=? 

2.  303  =  ?  5,      803  =  ?  8.    8003  =  ? 

3.  503=?  6^    2003=?  9.    9003=? 

431.  The  cube  of  a  number  composed  of  tens  and  units 
may  be  found  as  follows : 

24  =  20  +  4  =  2  tens  +  4  units. 

243  =  (-20  +  4)  X  (20  +  4)  X  (20  +  4). 

20  +  4   =        24 
20  +  4   =        24 


(20x4)  +  42zz:       96. 
202  +  (20  X  4)  =      480 


202  +  2  X  (20  X  4)  +  42  =:      576 
20  +  4   =        24 


(202  X  4)  +  2  X  (20  X  42)  +  4^  =    2304 
203  +  2  (202  X  4)  +  (20  X  42)  =    1152 


203  +  3  X  (202  X  4)  +  3  X  (20  x  42)  +  43  =  13824 

From  the  operation  we  find  that. 

The  cube  of  the  tens 203  =    8000 

3  times  the  square  of  tens  by  units 3  (202  x  4)  =    4800 

3  times  the  tens  by  the  square  of  the  units      .     .     3  (20  x  42)  =      960 

The  cube  of  the  units 43  =       64 

8000  +  4800  +  960  +  64  =  13824 

432.  Principle.  —  The  cube  of  a  number  composed  of 
tens  and  units  is  equal  to  the  cube  of  the  tens  plus  3  times 
the  square  of  the  tens  by  the  units,  plus  3  times  the  tens 
by  the  square  of  the  units,  plus  the  cube  of  the  units. 


320  EVOLUTION. 

Formula.  —  Tens'  +  3  x  tens^  x  units  +  3  x  tens  x  units' 
+  units^. 

10.  Separate  the  following  into  tens  and  units,  and  find 
their  cubes :  35,  54,  63. 

433.  By  reversing  the  process,  we  may  find  the  cube 
root. 

11.  What  is  the  cube  root  of  13824  ? 

Solution.  —  Separating  into  periods  of  three  figures  each,  begin- 
ning at  units,  we  have  13'824.  Since  there  are  two  periods  in  the 
power,  there  must  be  two  figures  in  the  root,  tens  and  units. 

The  greatest  cube  of  even  tens  contained  in  13824  is  8000,  and  its 
cube  root  is  20  (2  tens). 

13'824  I  20 +  4 

Tenss  =  20^  =  8000 

3  X  tens2  =  3  x  202  =  1200        5824 
3  X  tens  x  units  =  3  x  (20  x  4)  =    240 
units2  =  42  =      16 
3  X  tens2  +  3  tens  x  units  +  units2  =  1456 
(3  X  tens2  +  3  x  tens  x  units  +  units2)  x  units  =  5824 

Subtracting  the  cube  of  the  tens,  8000,  the  remainder,  5824,  con- 
sists of  3  X  (tens2  x  units)  +  3  x  (tens  x  units2)  -f  units^.  5824,  there- 
fore, is  composed  of  two  factors,  units  being  one  of  them,  and  3  x  tens^ 
+  3  X  tens  x  units  +  units^,  being  tlie  other.  But  the  greater  part  of 
this  factor  is  3  x  tens2.  By  trial  we  divide  5824  by  3  x  tens2  (1200) 
to  find  the  other  factor  (units) ,  which  is  4  if  correct.  Completing  the 
divisor,  we  have  12002  +  3  x  (20  +  4)  +  42  =  1456,  which,  multiplied 
by  the  units,  4,  gives  the  product,  5824,  proving  the  correctness  of  the 
work.     Therefore,  the  cube  root  is  20  +  4  =  24. 

434.  To  find  the  cube  root  by  the  aid  of  blocks. 

Finding  the  cube  root  of  a  number  is  equivalent  to  find- 
ing the  thickness  of  a  cube,  its  volume  being  given. 

The  following  formulas  illustrate  the  principles  that 
underlie  operations  in  cube  root. 


CUBE  ROOT. 


321 


Note. — For  convenience,    Z,   6,    «,    and  v  will  represent  length, 
breadth,  thickness,  and  volume,  respectively. 

(l)lxbxt  =  v.    (2)v^(lxh)^t.     (3)  v-^(lxt)  =  b. 
(4)  v---(bxt)  =  l 

12.    What  is  the  thickness  of  a  cube  whose  volume  is 
13824  cubic  feet  ? 


PRODUCT  OF  LENGTH 
AND  BREADTH. 

VOLUMES. 

THICKN 

3  X  202          =  1200 

13'824 

20  ft. 

3  X  20    X  4  =    240 

8000 

4  ft. 

42=      16 
1456 

5824 
5824 

24  ft. 

Solution.  —  The  greatest 
cube  of  even  tens  contained 
in  13824  cu.  ft.  is  8000  cu.  ft. 
(Cube  A.)  Its  thickness, 
therefore,  is  20  ft.  Sub- 
tracting 8000  (A)  from  13824  leaves  a  remainder  of  5824  cu.  ft., 
which  are  added  in  solids  of  equal  thickness  to  three  sides  of  A,  as 


^ 


Fig.  2. 


e 


'^ 


l^ 

— ^ 

e 

^ — 

1 

/ 

^— 

1 

9 

seen  in  Fig.  2.  It  now  remains  to  find  the  thickness  of  the  additions 
(6,  r,  d),  (e,  /,  g),  and  h,  which  have  a  uniform  thickness.  As  the 
solids  h^  c,  d  form  the  greater  part  of  the  volume  of  the  additions 
(5824  cu,  ft.),  and  the  length  and  breadth  of  each  is  20  ft.  (the  length 


322  EVOLUTION. 

and  breadth  of  A),  hj  trial,  using  Formula  2,  we  find  5824-^(3x202) 
=4  ft.,  thickness  of  the  additions,  if  correct.  Knowing  the  thickness, 
which  is  also  the  breadth  of  e,  /,  g,  h,  we  find  the  product  of  the 
length  and  breadth  of  e,  /,  gr  =  3  x  20  x  4  =  240  sq.  ft. ;  and  that  of 
/i  =  42  =  16  sq.  ft. ;  both  of  which  added  to  1200  sq.  ft.  =  the  product 
of  the  length  and  breadth  of  all  the  additions.  This  product,  by  For- 
mula 1,  multiplied  by  the  thickness,  4  ft,  =  5824  cu.  ft.,  proving  the 
correctness.     Therefore, 

The  thickness  of  a  cube  whose  volume  is  13824  cu.  ft.  is  20  +  4  ft. 
=  24  ft. 

The  numbers  in  the  middle  column  (Ex.  12)  all  indicate  volume  : 
13824  =  volume  of  original  cube. 
8000  =  volume  of  Cube  A. 
5824  =  volume  of  the  additions  (6,  c,  c?),  (e,  /,  g),  and  h. 

The  numbers  in  the  left-hand  column  indicate  product  of  length 
and  breadth : 

12JJ0  =  1  xb  of  solids  &,  c,  d. 

240  =  Z  X  &  of  solids  e,  /,  g. 
16  =  1  X  b  ot  cube  h. 

The  numbers  in  the  right-hand  column  indicate  thickness  : 
20  ft.  =  thicknels  of  A. 
4  ft.  =  thickness  of  all  the  additions. 
24  ft.  =  thickness  of  original  cube. 

435.    Short  method. 

Rule  for  finding  the  cube  root: 

Beginning  at  the  decimal  point,  separate  the  number  into 

periods  of  three  figures  each  ;  thus:  16'581'.375. 
Find  the  greatest  cube  in  the  left-hand  period,  and  write 
its  root  at  the  right.      Subtract  the  cube  from  the  left- 
hand  period,   and   bring   down   the  next  period  for  a 
dividend;  thus: 

16'581'.375  12  - 
8 
8581 


CUBE  KOOT. 


323 


To  find  the  trial  divisor,  square   the   root   already  found 
with  a  cipher  annexed,  and  multiply  by  3;  thus: 

16'581'.376  L2 
8  20 


Trial  divisor,  1200)8581 


_20 
400 

3 

1200 


3 


To  find  the  trial  figure,  find   how   many  times  the   trial 
divisor  is  contained  in  the  dividend;  thus:  ^ 


16'581'.375  |_25 
8  20 


Trial  divisor,  1200  )  8581 


20 
400 

3 

1200 


n 


n 


To  find  the  correction,  multiply  the  former  root  by  3,  an- 
nex the  trial  figure,  and  multiply   by   the   trial  figure; 

thus : 

16'581'.375  125.5 


8 

2 
3 

1200 

8581 

325 
Complete  divisor,  1525 

7625 

65 

5 

325 

187500 

956375 

Continue  thus,  until 

3775 

all  the  periods  are  ex- 

191275 

956375 

hausted. 

Note  1.  —  When  there  is  a  remainder  after  all  the  periods  are 
exhausted,  annex  decimal  periods,  and  continue  the  process  as  far 
as  desired.      The  result  will  be  the  approximate  root. 

Note  2.  —  When  a  cipher  occurs  in  the  root,  we  annex  two 
ciphers  to  the  trial  divisor,  and  bring  down  the  next  period. 

Note  3,  —  The  right-hand  decimal  period  must  have  three 
places. 


324  EVOLUTION. 

13.   What  is  the  cube  root  of  8.414975304? 
Operation. 
8.414'975'304  [  2.034 

I      ^ 

Since  0  occurs  in  the  root,  an- 
nex 00  to  the  trial  divisor,  mak- 
ing 120000 ;  bring  down  the  next 
period. 


120000 

1809 

121809 

414975 
365427 

6362700 

24376 

12387076 

49548304 
49548304 

Note.  —  To  find  the  cube  root  of  a  common  fraction,  extract  the 
root  of  each  term  separately.  If  both  terms  are  not  cubes,  reduce  to 
a  decimal  and  then  extract  the  root.  The  result  will  be  the  approxi- 
mate root. 

Find  the  cube  root  of :  ^  ^ 

14.  42875        '•''  19.  17.373979   ^  "X*^ 

15.  884736   *    ^    ,^  20.  450827        ^76  + 

16.  4492125   ^  ^^"^  21.  1879.080904     I'^'^H-^^ 

17.  77854483  "^   ^^^  22.  32.890033664  5,7^^i^ 

18.  8.615125   zP^^  23.  10077696       -z.  %\J^   "^ 

24.  What  is  the  cube  root  of  |||f|i?  -f^?  xlk?  39^J, 

Extract  the  cube  root  to  the  third  dermal  place : 

25.  14.323  i^*^^^*    27.    .06324i.i.^2  29.    3-1*^ 

26.  31982.4- -J^.X J'*'*'  28.    .0015   -^^^  ^         30.    7    »     »'^ 

31.  What  is  the  width  of  a  cube  whose  solidity  is  91125 
cubic  inches  ?     i  M  ^ 

32.  A  cubical  cistern  holds  50  barrels  of  water.  How 
deep  is  it? 

33.  What  is  the  entire  surface  of  a  cube  whose  side  is 
9  ft.? 

3> 


34.    ■</.006  X  32.5  =  ?  ^1^* 


SIMILAR   SOLIDS.  325 

SIMILAR   SOLIDS. 

436.  Solids  having  the  same  form  without  regard  to  size 
are  Similar  Solids.  Any  two  cubes  or  any  two  spheres  are 
similar  solids.  Solids  are  similar  when  their  correspond- 
ing dimensioi^  are  proportional. 

Principles.  —  Similar  solids  are  to  each  other  as  the 
cubes  of  their  corresponding  dimensions. 

The  corresponding  dimensions  of  similar  solids  are  to 
each  other  as  the  cube  roots  of  their  volumes. 

1.  A  globe  is  3  inches  in  diameter,  and  another  6  inches 
in  diameter.     What  is  the  ratio  of  their  volumes  ? 

Explanation.  —  They  are  to  each  other  as  3^  to  6^  =  27  :  216. 

2.  There  are  64  cubic  inches  in  a  4-inch  cube.  How 
many  in  an  8-inch  cube  ? 

3.  Two  similar  solids  contain  386  and  284  cubic  inches 
respectively.  If  the  larger  is  11  inches  thick,  how  thick 
is  the  smaller  ? 

4.  If  a  man  6  ft.  2  in.  tall  weighs  215  pounds,  what 
should  be  the  weight  of  a  man  5  ft.  10  in.  tall  of  the  same 
proportions  ? 

5.  The  width  of  a  bin  is  4  ft.  6  in.  How  wide  must  a 
similar  bin  be  to  hold  4  times  as  much  ? 

6.  If  an  orange  2^  inches  in  diameter  costs  5  cents, 
what  should  an  orange  3|-  inches  in  diameter  cost? 

QUESTIONS. 

437.  1.  What  is  involution?  A  power  of  a  number? 
The  first  power  ?  The  second  power  ?  The  third  power  ? 
What  are  the  second  and  third  powers  called  ?  Wliat  is 
the  exponent  of  a  power? 


326  EVOLUTIOK. 

2.  What  is  evolution  ?  A  root  ?  The  square  root  of  a 
number?  The  cube  root  of  a  number?  The  fourth  root 
of  a  number?  How  is  a  root  indicated?  The  square 
root  ?      The  fourth  root  ? 

3.  Tell  how  to  find  the  side  of  a  square  when  the  area 
is  given. 

4.  Tell  how  to  find  the  edge  of  a  cube  when  its  volume 
is  given. 

5.  What  kind  of  a  measure  is  a  cube  ? 

6.  A  cube  contains  how  many  times  as  many  figures  as 
its  root  ? 

What    is    shown   when    the   number   is    separated    into 
periods  of  three  figures  each? 

7.  What  is  the  cube  root  of  a  number  ?     Two  answers. 

8.  Cube  the  numbers  from  1  to  10. 

9.  What  is  the  first  root  figure  ?  What  kind  of  meas- 
ure is  it  ? 

10.  How  is  the  trial  divisor  found?     What  is  the  trial 
divisor  ? 

11.  What  kind  of  measure  is  it?      Why  is  it  a  trial 
divisor  ? 

12.  How  is  the  correction  found  ? 

13.  What  kind  of  measure  is  the  correction  ? 

14.  What  is  the  complete  divisor  ?     What  kind  of  meas- 
ure is  it  ? 

15.  What  is  a  right-angled  triangle? 

16.  What   principles    are   true    of   all   right-angled   tri- 
angles ? 

17.  Tell   how  to  find  hypothenuse,  base,  perpendicular. 

18.  What   are    similar   figures?      What    principles    are 
true  of  them  ? 

19.  What  are  similar  solids  ?     What  principles  are  true 
of  them? 


GENERAL   REVIEW. 


438.   Oral. 

1.  What  is  the  cost  of  20  pounds  of  sugar  at  6|-  cents 
a  pound? 

2.  A  man  owning  |  of  a  farm  sold  ^  of  his  share. 
What  part  does  he  still  own? 

3.  A  can  do  a  piece  of  work  in  2  hours,  and  B  in  3 
hours.     In  what  time  can  both  do  it,  working  together  ? 

4.  Two  men  receive  #  60  for  painting  a  house.  One 
worked  for  ^  2  a  day,  and  the  other  ^  3  a  day.  How  much 
money  should  each  receive  ? 

5.  What  is  the  interest  of  $500  for  2^-  years  at  6%  ? 

6.  What  is  the  cost  of  64  straw  hats  at  $  1  each  ?     At 

$.50?     $.25?     At$.12i?     At  $1.25?     At  $2.50? 

7.  If  4  oranges  cost  12  cents,  what  will  7  oranges  cost  ? 

8.  If  f  of  a  yard  of  silk  costs  $  1|,  what  will  1^  yards 
cost? 

9.  If  a  man  6  feet  tall  casts  a  shadow  8  feet  long,  how 
long  a  shadow  will  a  boy  4i  feet  tall  cast  ? 

10.  If  I  of  my  money  is  silver  and  the  rest  bills,  and  I 
have  $  180,  how  much  of  each  kind  have  I  ? 

11.  If  f  of  a  cord  of  wood  costs  $  1.50,  what  will  a  cord 
cost  ?  5  cords  ? 

327 


S^8  GENERAL   REVIEW. 

12.  A  boy  buys  papers  at  the  rate  of  3  for  2  cents,  and 
sells  them  at  the  rate  of  2  for  5  cents.  How  much  does  he 
make  on  30  papers  ? 

13.  What  is  the  value  of  8  bushels  of  wheat,  if  6 
bushels  cost  $4.50? 

"     14.    What  is  the  cost  of  2  lb.  8  oz.  of  butter  at  16  cents 
a  pound  ? 

15.  What  is  the  difference  between  5  square  feet  and  5 
feet  square  ? 

16.  When  it  is  noon  in  Boston,  what  time  is  it  7^°  east 
of  Boston  ? 

17.  Two  places  are  37|-  degrees  apart.  When  it  is  5 
P.M.  at  the  eastern  place,  what  is  the  time  of  the  western? 

18.  When  it  is  noon  in  Chicago,  what  is  the  time  60° 
west  of  Chicago  ? 

'     19.    What   is   the  standard  time  of   Denver  when  it   is 
noon  in  Boston  ? 

20.  58  is  -|  of  what  number  ? 

21.  A  boy  sold  a  knife  for  60  cents,  which  was  f  of  its 
cost.     What  did  it  cost  ? 

22.  The  sum  of  two  numbers  is  32;  their  difference  is 
10.     What  are  the  numbers  ? 

Note. — The  half-sum -f  the  half-difference  =  the  greater.  The 
half-sum  —  the  half-difference  =  the  less. 

23.  At  a  village  election  there  were  1200  votes  cast  for 
two  candidates ;  the  successful  candidate  had  a  majority  of 
200  votes.     How  many  votes  were  cast  for  each  ? 

24.  The  sum  of  two  numbers  is  6S;  their  difference  is 
26.     What  are  the  numbers  ? 

25.  What  is  33|-%  of  $900?  66f%  of  $1200?  12i%, 
of  $96?     25%  of  $600? 


PEOBLEMS.  329 

26.  A  merchant,  by  selling  goods  at  $80,  lost  20%. 
What  was  the  cost  ? 

27.  A  farmer  had  a  flock  of  sheep  and  purchased  25% 
more;  he  then  had  250  sheep.  How  many  had  he  at 
first? 

28.  A  lad  had  45  marbles  and  lost  33-J^%  of  them.  How 
many  had  he  left  ? 

29.  What  is  an  agent's  commission  for  buying  96  head 
of  cattle  at  $  33^  a  head,  at  6|%  ? 

30.  75  x66f -26x121  =  ? 

31.  How  much  is  500%  of  $  12  ? 

32.  A  druggist  expended  $20  in  opium,  which  he  sold 
at  a  profit  of  300%.     What  did  he  sell  it  for  ? 

33.  $18  is  600%  of  what  ? 

34.  What  is  the  difference  between  .6%  of  $50  and  |% 
of  $70? 

35.  What  per  cent  of  a  number  is  ^  of  it?  -J-  of  it? 
-^  of  it?  I  of  it ?  f  of  it?  I  of  it?     16  is  i%  of  what? 

36.  A  lot  containing  48  square  rods  is  3  times  as  long  as 
it  is  wide.     What  are  its  dimensions  ? 

Explanation.  — As  the  length  is  three  times  the  breadth,  we  divide 
the  area  by  3  ;  the  result  will  be  the  area  of  each  of  3  equal  squares, 
the  square  root  of  which  will  be  the  width,  which  multiplied  by  3 
will  give  the  length.     V^  =  4  rd.,  the  width. 

37.  A  and  B  had  the  same  income.  A  saved  \  of  his 
and  B  ^.  A  had  $  1600  at  the  end  of  8  years.  How  much 
had  B  ? 

38.  Which  is  greater,  the  square  root  of  ^\,  or  the  cube 
ofi? 

39.  A  two-inch  pipe  can  discharge  the  contents  of  a  cask 
in  8  hours.     How  long  will  it  take  a  four-inch  pipe  ? 


330  GENERAL  BEVIEW. 

40.  How  many  rods  of  fence  necessary  to  fence  a  square 
lot  containing  144  sq.  rd.  ? 

41.  A  lot  containing  144  sq.  rd.  is  four  times  as  long 
as  it  is  wide.  How  many  rods  of  fence  does  it  require? 
(Compare  with  the  result  in  Ex.  40.) 

42.  How  many  inches  in  a  hektometer? 

43.  How  many  milliliters  in  4  dekaliters  ? 

44.  How  many  ares  in  5  hektares  ? 

45.  John  and  George  divide  150  marbles  in  proportion 
to  their  ages.  John  is  7,  and  George  is  8.  How  many 
marbles  do  each  receive  ? 

46.  If  a  boy  can  ride  a  bicycle  at  the  rate  of  18  miles  an 
hour,  how  long  will  it  take  him  to  ride  twice  around  a  sec- 
tion of  land  ? 

47.  What  is  the  interest  of  $600  at  8%  for  3  months? 
For  3  years  ? 

43.  If  I  owe  a  debt  of  $60,  and  pay  $40  two  months 
before  it  is  due,  how  long  after  it  is  due  should  the  re- 
mainder be  allowed  to  run? 

49.  At  what  time  between  3  and  4  o'clock  are  the  hour 
and  the  minute  hand  of  a  watch  together  ? 

Explanation.  — Both  hands  are  together  at  12  o'clock,  and  before 
it  is  12  o'clock  again  they  will  have  been  together  11  times.  They  will 
be  together  between  1  and  2  in  -^j  of  12  hours,  and  between  3  and  4  in 
^  of  12  hours. 

50.  If  I  buy  8%  stock  so  that  it  pays  me  6%  on  my  in- 
vestment, what  per  cent  do  I  receive? 

Written. 

51.  Frost  injured  72  peach  trees  on  M's  farm,  which 
number  was  9%  of  all  the  trees  he  had.  How  many  did 
he  have  in  all  ? 


PROBLEMS.  331 

52.    At  2%  an  agent  received  $125.50  commission  on  the 
sale  of  some  real  estate.     What  was  it  sold  for  ? 


$150  Dubuque,  la.,  Jan.  1,  1896 

Three     months     after     date,    I    promise    to    pay 
Storrie  ^  Dunlap ^^^^^^^^^^^^^^^ov  order,  One  hun- 
dred fifty  Dollars,  with  interest.     Value  received. 

J.    W.  Kimhall. 

Find  the  proceeds  of  the  above  note,  discounted  at  the 
First  National  Bank,  Feb.  16,  1896. 

54.  A  gentleman  insured  his  house  for  $1800,  which 
was  f  of  its  value,  at  1\%.  In  case  of  total  destruction 
by  fire,  what  is  the  entire  loss  to  the  owner  ? 

55.  A  bill  of  goods  amounting  to  $287.60  is  sold  with 
discounts  of  10%  and  5%  for  cash.  How  much  cash  will 
pay  it  ? 

56.  If  a  piano  that  cost  $  360  is  to  be  sold  at  a  profit  of 
16|%,  what  price  must  be  asked  that  12^%  may  be  abated 
from  the  asking  price  ? 

57.  I  sold  two  articles  for  $  1.50  each,  thereby  realizing 
a  profit  of  25%  on  one  and  a  loss  of  25%  on  the  other. 
Did  I  gain  or  lose  on  both  transactions  ? 

58.  A  bought  a  carriage  at  20%  and  10%  from  list 
price,  and  sold  it  at  10%  and  5%  from  list  price.  What 
per  cent  profit  did  he  make  ? 

59.  A  grocer  bought  a  cask  of  molasses  containing  40 
gal.  for  38  cents  per  gallon.  Seven  gallons  having  leaked 
out,  for  how  much  per  gallon  must  he  sell  the  remainder  in 
order  to  gain  12^%  on  the  investment  ? 


332  GENERAL    REVIEW. 

60.  Suppose  a  grocer  bought  a  42-gallon  cask  of  vinegar 
at  12  ^  per  gallon,  and  put  12  gallons  of  water  with  it,  and 
sold  it  for  the  same  price.  What  would  be  his  rate  per 
cent  of  gain  ? 

61.  A  meter  stick  is  what  per  cent  longer  than  a  yard 
stick  ? 

62.  Buffalo  is  the  largest  flour  depot  in  the  world.  It 
received  by  lakes  and  rail  in  1895,  8,971,740  bbl.  of  flour. 
If  the  N.  Y.  C.  &  H.  R.R.  shipped  18.9%,  the  N.  Y.,  L.  E., 
&  W.  E.K  12.15%,  the  Pennsylvania  E.E.  8.33%,  the 
West  Shore  R.R.  10.97%,  the  Lehigh  Valley  E.E.  8.12%, 
the  other  roads  6.5%,  and  the  remainder  by  water,  what 
per  cent  was  shipped  by  water,  and  how  many  barrels  ? 

63.  What  will  be  the  cost  of  6  loads  of  wood,  each  con- 
taining 1  C.  6  cd.  ft.  10  cu.  ft.,  at  $  2.50  a  cord  ? 

64.  How  many  yards  of  carpet  2  ft.  wide  will  be  re- 
quired for  a  room  12  ft.  by  15  ft.  6  in.,  if  the  strips  run 
lengthwise,  and  there  is  a  waste  of  J  of  a  yard  in  each  strip 
in  matching  ? 

65.  The  width  of  a  building  is  36  ft.,  and  the  ridge  of 
the  roof  is  10  ft.  higher  than  the  eaves.  How  many  square 
feet  of  boards  will  it  take  to  cover  one  of  the  gable  ends  ? 

66.  With  how  long  a  rope  must  a  goat  be  fastened  to  a 
stake  that  it  may  feed  on  four  square  rods  of  land  ? 

67.  A  room  24  feet  long  and  15  feet  wide  is  to  be  car- 
peted with  carpet  |-  yd.  wide.  How  many  yards  will  be 
required  if  a  waste  of  ^  of  a  yard  is  made  on  each  strip  in 
matching,  the  strips  to  run  crosswise  ? 

68.  Oswego,  N.Y.,  is  in  latitude  43°  28'  N.  How  many 
degrees  is  it  from  the  North  Pole  ?     From  the  South  Pole  ? 

69.  How  many  gallons  in  32^-  hektoliters  of  wine  ? 


PROBLEMS. 

70.  If  it  takes  2  lb.  7  oz.  4  pwt.  of  silver  to  make  12 
spoons,  what  amount  will  be  required  for  one  spoon  ? 

71.  If  it  is  one-half  of  a  mile  from  your  home  to  the 
school  building,  how  many  steps  of  1  ft.  6  in.  each  will 
you  take  in  reaching  it  ? 

72.  What  decimal  part  of  a  week  is  4  da.  3  hr.  36  min.  ? 

73.  What  part  of  2  reams  are  10  quires,  20  sheets  ? 

74.  How  many  times  is  132  x  75  x  42  x  104  contained 
in  26  X  22  X  150  x  168? 

75.  Bought  six  loads  of  oats,  each  containing  32  bags, 
each  bag  containing  2  bushels,  worth  $  .56  a  bushel,  and 
gave  in  return  8  boxes  of  tea,  each  containing  24  pounds. 
What  was  the  tea  worth  a  pound  ? 

^g    4  X  7  X  32  X  15  X  88  _  ^ 
16x56x5x4x6 

77.  If  f  of  a  box  of  oranges  cost  $  4.50,  what  part  of  a 
box  can  be  bought  for  ^  5.25  ? 

78.  Simplify  the  following  complex  fraction: 

4    V    s         5     I     3 


79.  I  of  63  is  j\  of  what  number  ? 

80.  A  gentleman  invested  $  215380  in  a  knitting-mill, 
which  was  f  of  the  value  of  the  plant.  What  was  the 
value  of  ^  of  the  plant  ? 

81.  A  and  B,  being  150  miles  apart,  travel  toward  each 
other.  They  start  at  the  same  time,  and  meet  at  the  end 
of  eight  hours,  when  they  discover  that  A  has  travelled  1^ 
miles  each  hour  more  than  B.  How  many  miles  has  each 
man  travelled  ? 


334  GENERAL   REVIEW. 

82.  For  how  long  a  time  must  $  4560  be  placed  on  inter- 
est at  6%  to  gain  $353.40? 

83.  A  man  borrowed  $  250  March  3,  1896,  and  paid  the 
note  Sept.  21,  1896,  with  5%  interest.  What  was  the 
amount  of  the  note  ? 

84.  A  merchant  borrowed  f  165  at  6%,  and  when  he 
paid  the  debt  it  amounted  to  $  168.96.  How  long  did  he 
have  the  use  of  the  money  ? 

85.  The  interest  on  a  certain  sum  is  $  27.40,  the  time  2 
years,  3  months,  12  days,  and  the  rate  6%.  What  is  the 
principal  ? 

86.  A  note  for  $  250  was  given  Sept.  5, 1895.  A  payment 
of  $  75  was  made  April  25,  1896.  How  much  will  settle 
the  note  Oct.  3,  1897,  interest  at  6%  ? 

87.  A  man  bought  a  farm  for  $  4000,  April  1,  1889.  He 
gave  a  mortgage  at  5%  for  $3000,  and  paid  as  follows: 
Jan.  1,  1890,  $  700 ;  Oct.  1,  1890,  $  1000 ;  April  1,  1891, 
$850;  and  the  balance  of  the  mortgage  April  1,  1892. 
How  much  was  due  at  settlement  ? 

88.  What  sum  of  money  must  I  loan  at  6  per  cent  inter- 
est, that  it  may  bring  me  in  a  quarterly  income  of  $  300  ? 

89.  Compute  the  interest  on  $  3450  for  2  yr.  6  mo.  20  da. 

at  5%. 

90.  William  Johnson  holds  a  note  for  $1250  against 
James  W.  Way,  dated  Jan.  10,  1893,  payable  on  demand. 
This  note  bears  the  following  indorsements :  March  10, 
1893,  $200;  May  10,  1893,  $300;  July  10,  1893,  $50; 
Oct.  10,  1893,  $400.  What  is  due  Dec.  10,  1893,  interest 
at  5%  ? 

91.  Find  the  simple  interest  of  $382.94,  one  half  to  be 
paid  in  5  yr.  5  mo.  20  days  at  3%,  the  other  half  to  be  paid 
in  5  yr.  5  mo.  20  days  at  5%. 


PROBLEMS.  335 

92.  A  man  borrows  $  2000  which  belongs  to  a  minor 
who  is  18  yr.  2  mo.  10  days  old,  and  he  is  to  keep  it  until 
the  owner  is  21  years  of  age.  What  will  then  be  due, 
money  being  worth  6%  ? 

93.  Bought  a  house  for  ^6000,  and  gave  a  mortgage  for 
$4000,  dated  Jan.  1,  1892,  interest  at  6%.  Made  the  fol- 
lowing payments  :  July  1,  1892,  $  520 ;  Jan.  1,  1893,  $  708  ; 
Jan.  1,  1894,  $680;  July  1,  1895,  $725.  How  much  was 
due  Jan.  1,  1896  ? 

94.  A  man  owes  me  $  463.50,  payable  in  6  months  with- 
out interest.  What  sum  can  I  afford  to  take  now  for  the 
debt,  money  being  worth  6%  ? 

95.  A  man  bought  goods  amounting  to  $  2100  on  6  mo. 
credit,  but  was  offered  a  discount  of  3%  cash  payment. 
If  money  was  worth  ^%  a  month,  what  is  the  difference  ? 

96.  Which  is  the  more  profitable,  to  buy  goods  worth 
$500  at  90  days,  3%  off  for  cash,  or  put  the  amount  at 
interest  at  7%,  and  let  the  bill  run  to  maturity  ? 

97.  Face  of  a  debt,  $1256.25.  Date,  July  1,  1886. 
Time,  1  yr.  6  mo.     Eate,  6%.     What  is  the  present  worth  ? 

98.  Had  a  note  of  $  2500  discounted  at  a  Milwaukee  bank 
for  two  months.  What  were  the  proceeds,  rate  of  discount 
being  7%  ? 

99.  Find  the  difference  between  the  true  discount  and 
the  bank  discount  of  a  debt  of  $  550,  due  in  4  months 
without  interest. 

100.  April  1,  A  gave  B  a  3-mo.  note  for  $  300,  which  B 
had  discounted  at  a  bank  May  1.  What  did  B  receive, 
and  what  amount  could  the  bank  collect  on  July  1,  dis- 
count at  6%,  no  grace? 

101.  Bought  an  invoice  of  goods  amounting  to  $  1360.58. 
How  much  will  I  make  by  discounting  my  note  at  the  bank 
for  90  days  at  6^,  and  paying  cash  for  goods  at  5%  off'  ? 


336  GENERAL   REVIEW. 

102.  A  New  York  note  of  $  2000,  bearing  date  May  24, 
1895,  and  payable  in  60  days,  was  discounted  at  6%.  The 
discount  was  $  15.     When  was  it  discounted  ? 

103.  Sweet  and  Johonnot  sold  20  bicycles  to  a  dealer, 
taking  his  note  at  60  days,  which  they  discounted  immedi- 
ately at  the  Merchant's  Bank  at  6%  with  grace,  receiving 
$  1485.     What  was  the  price  of  each  bicycle  ? 

104.  On  the  first  day  of  January,  1890,  a  man  gave  three 
notes,  the  first  for  $  500  payable  in  30  days ;  the  second  for 
$  400  payable  in  60  days ;  and  the  third  for  $  600  payable 
in  90  days.  What  was  the  average  term  of  credit,  and 
what  was  the  equated  time  of  payment? 

105.  I  wish  to  use  $  560.88  immediately.  For  what  sum 
must  I  draw  a  bank  note,  due  in  96  days  at  6%,  that  I  may 
receive  the  required  amount  ? 

106.  How  many  $500  U.  S.  bonds  can  be  bought  for 
$  6630  at  101%  premium  ? 

107.  A  guardian  invests  $  1000  at  simple  interest  at  3%, 
$1000  in  4%  bonds  at  1121,  and  $1000  in  5%  bonds  at 
125.  The  bonds  are  to  run  10  years,  and  be  redeemed  at 
par.  Compare  the  three  investments  at  the  end  of  the  ten 
years. 

108.  The  city  of  Buffalo  pays  $  12425.72  for  rented  school 
buildings.  On  what  amount  of  3|-%  bonds  would  this  pay 
the  interest  ? 

109.  If  I  buy  bank  stock  at  20%  discount,  and  sell  it  at 
10%  premium,  what  per  cent  do  I  gain  ? 

110.  What  is  the  rate  of  income  on  a  4%  stock  bought  at 
62  J? 

111.  I  have  $5000  to  invest,  and  can  buy  5%  stock  at 
110,  or  6%  stock  at  125.  Which  will  be  the  better  invest- 
ment, and  how  much  annually  ? 


PBOBLEMS.  337 

112.  A  gentleman  owned  a  house  which  he  rented  for 
$375  above  all  expenses.  He  sold  the  house  for  $5000, 
and  invested  the  money  in  a  5%  stock  at  80.  Did  he  gain 
or  lose  by  the  transaction,  and  how  much  per  year  ? 

113.  The  ratio  of  A's  weight  to  that  of  B  is  f .  B  weighs 
120  lb.  8  oz.     What  does  A  weigh  ? 

114.  If  John  is  6  years  old  and  Henry  15,  what  is  the 
ratio  of  John's  age  to  that  of  Henry?  What  will  it  be 
when  each  is  5  years  older  ? 

115.  If  4  horses  eat  4  bu.  of  oats  in  2  days,  how  many 
horses  will  eat  48  bushels  in  12  days  ?     (Solve  by  analysis.) 

116.  If  the  antecedent  is  |  of  -f^  x  ^%,  and  the  ratio  is 
-|  of  Yi)  what  is  the  consequent  ? 

117.  How  wide  can  20  men,  working  8  hours  a  day  for  8 
days,  make  a  ditch  which  is  75  rods  long  and  10  ft.  deep,  if 
25  men,  working  10  hours  a  day  for  7  days,  can  dig  a  ditch 
80  rd.  long,  8  ft.  deep,  and  2  ft.  wide  ? 

118.  If  a  baker's  loaf  weighs  10  ounces  when  wheat  is  60 
cents  a  bushel,  what  should  it  weigh  when  wheat  is  70  cents 
a  bushel  ? 

119.  If  a  train  moves  at  the  rate  of  30  miles  in  48 
minutes,  in  what  time  will  it  run  450  miles  ? 

120.  One  side  of  a  shed  is  8  ft.  high,  the  opposite  side  13 
ft.  6  in.     What  is  the  ratio  between  the  sides  ? 

121.  If  it  costs  $30  to  lay  a  cement  sidewalk  4  ft.  wide 
and  16  ft.  long,  how  much  will  it  cost  to  lay  the  same 
kind  of  walk  7  ft.  wide  and  96^  ft.  long  at  the  same  rate  ? 

122.  Write  and  solve  a  problem  in  proportion,  using  the* 
following  numbers ;  8  men,  8  lb.  of  beef,  1  da.  j  and  2  da., 
12  lb.  of  beef. 


3SS  GENERAL  REVIEW. 

123.  If  ^  of  a  yard  of  cloth  cost  $^,  what  will  4^  yd. 
cost  ? 

124.  A,  B,  and  C,  engaged  in  trade.  A  put  in  $400,  B 
$250,  C  $600.  They  gain  $300.  Find  each  man's  share 
of  the  gain. 

125.  A  merchant  failing  in  trade  has  debts  amounting  to 
$  34560 ;  his  assets  are  $  30240.  What  can  he  pay  on  the 
dollar,  and  how  much  will  a  creditor  receive  to  whom  he 
owes  $3840? 

126.  A  man  willed  his  property,  which  was  valued  at 
$6000,  to  his  four  children  in  the  following  proportion, 
giving  to  each  one  i,  J,  ^,  and  i  respectively.  How  much 
did  each  one  receive  ? 

127.  Three  families  rent  a  cottage  for  the  summer.  The 
■first  family  occupies  it  for  6  weeks,  the  second  for  2,  and  the 
third  for  3.  The  rent  for  the  entire  season  of  11  weeks 
is  $  440.     How  much  should  each  family  pay  ? 

128.  Scrantom,  Morris,  and  Jackson  were  associated  in 
business  for  a  period  of  1  yr.  6  mo.  Scrantom  furnished 
$5000,  Morris  $3000,  and  Jackson  $2000  of  the  original 
capital.  When  the  partnership  terminated,  they  divided 
$  4000,  the  profits  arising  from  the  same.  How  much  more 
did  each  make  than  he  would  have  realized  had  his  money 
been  invested  in  a  6%  mortgage  ? 

129.  Divide  $60  among  three  boys,  so  that  one  shall  have 
^  as  much  as  the  other  two,  whose  shares  are  as  3  to  7. 

130.  What  is  the  distance  between  the  diagonally  oppo- 
site corners  of  a  lot  whose  area  is  16  sq.  ft.  ? 

131.  My  dining  room  is  16  ft.  long,  14  ft.  wide,  10  ft. 
high.  Find  diagonals  of  the  shorter  sides,  of  the  longer 
sides,  and  of  the  room. 


PROBLEMS.  339 

1S2.  What  is  the  length  of  one  side  of  a  cube,  equal  in 
volume  to  a  solid  that  is  49  ft.  long,  27  ft.  wide,  and  7  ft. 
high  ? 

133.  A  ladder  25  ft.  long,  the  bottom  of  which  is  5  ft. 
from  a  building,  reaches  the  base  of  a  window.  How  many 
feet  from  the  base  of  the  window  to  the  ground  ? 

134.  At  40  cents  a  rod  for  fencing,  which  will  cost  the 
more,  to  enclose  a  square  field  containing  10  A.,  or  a  field  of 
the  same  area  whose  length  is  twice  its  width  ? 

135.  Find  the  cube  root  of  41781.923. 

136.  Find  the  square  root  of  41781923. 

137.  A  cubical  cistern  contains  30  hhd.  of  water.  How 
deep  is  it  ? 

"  138.  The  volume  of  a  rectangular  prism  is  200  cu.  ft., 
and  its  height  is  8  ft.  Find  its  surface  contents,  if  its  two 
other  dimensions  are  equal. 

139.  The  area  of  a  right-angled  triangle  is  289  sq.  ft.,  its 
base  is  ^  of  its  altitude.    What  is  the  length  of  its  altitude  ? 

140.  If  a  railroad  company  pays  19^  per  sq.  yd.  for 
excavating,  and  37^^  per  sq.  yd.  for  drawing  away  the 
earth,  what  will  it  cost  the  company  to  remove  a  mound 
equal  in  volume  to  a  cube  whose  side  is  81  feet  ? 

141.  Forty  feet  directly  east  from  a  column  that  is  75 
ft.  high,  I  measure  due  north  30  ft.,  and  find  that  I  am  in 
line  with  a  stake  and  the  column.  If  the  stake  is  25  ft. 
distant  from  my  position,  and  10  ft.  high,  what  is  the  dis- 
tance from  the  top  of  the  stake  to  the  top  of  the  column  ? 

142.  How  many  rods  of  fence  will  enclose  a  rectangular 
field  containing  20  acres,  if  the  field  is  twice  as  long  as  it 
is  wide,  and  how  much  will  it  cost  at  f  2.45  per  rod  ? 


340  TOPICAL  EEVIEW. 

143.  If  a  locomotive  runs  at  the  rate  of  55  miles  in  40 
minutes,  and  its  drive-wheels  are  18  ft.  in  circumference, 
how  many  revolutions  will  the  drive-wheels  make  in  one 
hour? 

144.  A  insured  his  stock  for  $  1200.  He  paid  a  premium 
of  $  24.     What  was  the  rate  of  insurance  ? 

146.  Grant  and  Dunn  bought  a  bill  of  glass  amounting 
to  $853.68,  upon  which  they  received  a  discount  of  60%, 
25%,  15%,  and  2%  off  for  cash.  What  was  the  net  amount 
of  the  bill? 

146.  My  agent  in  Chicago  sold  goods  to  the  amount  of 
$  8640.  He  also  purchased  6800  bu.  of  wheat  at  $  1.10  a 
bushel,  paid  for  expenses  $  10.40,  and  received  a  commis- 
sion of  2  ct.  on  every  dollar.  How  much  will  he  remit  to 
me  after  paying  all  expenses  ? 

147.  A  merchant  buys  calico  at  5|-  ct.  per  yard,  and  sells 
at  6.     What  is  his  rate  per  cent  of  gain  ? 

148.  What  is  the  rate  of  insurance  when  a  $  1000  policy 
for  3  years  costs  $  7.50  ? 

149.  A  man  lost  $  13.45  on  some  flour  by  selling  it  at  a 
loss  of  14^%.     What  was  the  flour  worth  ? 

150.  A  farmer  buys  4  tons  of  hay  at  $  20  per  ton,  and 
4  bbl.  of  flour  at  $  5  per  barrel.  What  is  the  cash  value 
of  the  bill,  if  he  is  allowed  a  discount  of  15%,  and  5% 
deduction  for  cash  ? 

TOPICAL   REVIEW. 

439.  Arranged,  by  permission,  from  examinations  given 
in  various  cities. 

1.  Define  Addition,  Sum,  Sign  of  Equality,  Subtraction, 
Remainder,  Subtrahend,  Minuend,  Parenthesis,  Multiplica- 
tion, Factors,  Multiplicand. 


NOTATION   AND   NUMERATION.  341 

2.  Multiply  7258  by  395,  and  write  each  partial  product 
in  words. 

3.  Subtract  8969  from  9782,  and  prove  the  work. 

4.  Prove  by  an  illustration  that  multiplication  resembles 
addition. 

5.  Solve :  $  73.46  -  ($.94  +  $  3.02)  +  $  47  x  35. 

6.  Write  in  figures,  XLVII. 

7.  Write  in  figures,  six  hundred  eight  thousand  seventy- 
two. 

8.  Multiply  6504  by  657. 

9.  f  of  585x5  =  ? 

10.    Divide  45897  by  490,  and  prove  that  your  work  is 
correct. 

440.    1.    Copy   and    find    the    sum:     $23.17,    $6043.05, 
$0.42,  $208.97,  $5486.04. 

2.  How  many  yards  of  linen,  at  28  cents  a  yard,  must 
be  given  for  35  bushels  of  potatoes,  at  56  cents  per  bushel  ? 

3.  A  man  paid  $  13,465  for  a  house  and  some  land. 
The  house  alone  was  worth  $  8978.  What  was  the  value 
of  the  land  ? 

4.  Write  this  number  in  words,  3,782,013. 

5.  Write  in  words,  XCV. ;  76508.904. 

6.  How  many  bushels  of  potatoes  at  50  cents  a  bushel 
will  pay  the  entire  cost  of  a  hat  at  $  7.50,  a  dress  at  $  24, 
a  cloak  at  $  16.25,  and  gloves  at  $  1.75  ? 

7.  6460000  X  3000  -  25000000  =  ? 

8.  If  the  dividend  is  -1761184  and  the  quotient  4684, 
what  is  the  divisor? 

9.  What  is  the  smallest  number  that  will  exactly  con- 
tain 16,  20,  24,  or  30  ? 

10.    Define   Dividend,   Kemainder,   Product,   the    Prime 
Factors  of  a  number.     How  do  you  prove  division  ? 


342  TOPICAL  REVIEW. 

441.  1.   Define  Multiplier ;  Concrete  Number. 

2.  How  can  you  prove  an  example  in  subtraction  ? 

3.  A  merchant  bought  375  bbl.  of  apples  at  $  .95  a  bbl. 
43  bbl.  rotted.  If  he  sells  the  rest  at  $  1.10  per  bbl.,  how 
much  does  he  gain  or  lose  on  all  ? 

4.  My  salary  is  ^2350  a  year,  and  I  spend  $4  a  day. 
How  much  will  I  save  in  six  years  ? 

5.  If  23  men  own  475  bbl.  of  apples  each,  and  4  of  them 
divide  theirs  equally  among  the  rest,  how  many  will  each 
have  then  ? 

6.  Find  the  prime  factors  of  1155. 

7.  If  my  salary  is  f  1400  per  year,  and  my  expenses 
$  90  per  month,  how  long  will  it  take  me  to  save  $  4160  ? 

8.  What  is  the  smallest  quantity  of  grain  that  will  fill 
an  exact  number  of  bins,  whether  they  hold  312,  260,  or 
390  bushels  ? 

9.  What  are  like  numbers  ?     Give  three. 

10.  From  Albany  to  West  Troy  is  5  miles,  from  West 
Troy  to  Cohoes  2  miles,  and  from  Cohoes  to  Saratoga  is  30 
miles.     How  far  is  it  from  Albany  to  Saratoga  ? 

442.  1.    Write  in  words,  23456789. 

2.  Find  the  greatest  common  divisor  of  75,  25,  and  500, 
and  their  least  common  multiple. 

3.  If  7  tons  of  hay  cost  $105,  what  will  be  the  cost  of 
289  tons  ? 

4.  Write  the  number  which  is  composed  of  3  units  of 
the  eighth  order,  6  of  the  fifth,  2  of  the  third,  and  9  of  the 
second. 

5.  Find  the  contents  of  the  smallest  measure  that  may 
be  filled  by  using  either  a  4-quart,  a  5-quart,  or  a  6-quart 
measure. 


FACTORING.  343 

6.  Find  the  prime  factor  of  1452. 

7.  Solve  by  cancellation:  A  man  receives  $21  for  15 
days'  work  of  7  hours  each.  How  much  should  he  receive 
for  19  days'  work  of  5  hours  each  ? 

8.  The  product  of  three  numbers  is  105840 ;  one  of  the 
numbers  is  42,  the  other  35.     What  is  the  third  number  ? 

9.  How  many  pounds  of  butter  at  20^  a  pound  are 
worth  as  much  as  1600  bushels  of  wheat  at  75  ^  a  bushel  ? 

10.  What  is  the  greatest  common  divisor  of  two  or  more 
numbers  ? 

443.  1.  Two  persons  start  from  the  same  point  and 
travel  in  opposite  directions;  one  at  the  rate  of  25  miles 
a  day,  and  the  other  at  the  rate  of  32  miles  a  day.  How 
far  apart  will  they  be  in  8  days  ? 

2.  What  is  the  product  of  20202  x  10101  ? 

3.  What  number  multiplied  by  1728  will  produce 
1705536? 

4.  A  man  has  $  8250.  How  much  must  he  add  to  this 
to  be  able  to  pay  for  a  farm  worth  $  10000  ? 

5.  (6070  -  1200)  +  (4680  -- 15)  =  ? 

6.  Bought  144  acres  of  land  at  $  41.25  an  acre,  and  sold 
the  whole  for  $  7000.     Did  I  gain  or  lose,  and  how  much  ? 

7.  If  3  oranges  are  worth  -^  of  a  melon,  what  part  of  the 
melon  is  1  orange  worth  ? 

8.  Austin  having  30  marbles,  gave  ^  of  them  to  one 
companion  and  ^  of  them  to  another.  How  many  had  he 
left? 

9.  How  many  eggs  in  12|^  dozen  ? 

10.   Write  the  present  year  in  Eoman  numerals. 


344  TOPICAL   REVIEW. 

444.  1.  George  gave  a  beggar  9  cents,  which  was  i  of  all 
the  money  he  had.     How  much  money  had  he  ? 

2.  Mary  is  14  years  old,  and  her  sister  is  f  as  old. 
How  old  is  her  sister  ? 

3.  What  is  a  mixed  number  ? 

4.  How  many  ninths  in  5^  ? 

5.  At  -f  of  a  dollar  a  pound,  what  will  8  pounds  of 
butter  cost  ? 

6.  What  will  f  of  a  pound  of  coffee  cost  at  28  cents  a 
pound  ? 

7.  If  a  man  earns  $  15  a  week  and  spends  f  of  it,  how 
much  does  he  save  ? 

8.  What  do  you  understand  by  |-  of  anything  ? 

9.  Change  1^2^  to  an  improper  fraction. 

10.  A  boy  having  20  quarts  of  blueberries,  sold  |  of  them 
for  $  -^-Q.     What  was  the  price  for  a  quart  ? 

445.  1.  If  I  put  i  of  my  money  in  one  bank,  i  in 
another,  ^  in  another,  and  have  $  4200  besides,  how  much 
have  I  ? 

2.  A  can  mow  a  field  in  10  days,  B  in  8  days,  and  C  in 
5  days.  When  working  together,  in  how  many  days  will 
they  all  do  it? 

3.  If  6  is  added  to  both  terms  of  the  fraction  ^,  how 
much  is  the  fraction  increased  or  diminished  ? 

4.  The  divisor  is  46,  the  quotient  60*5,  and  the  remainder 
23.     What  is  the  dividend  ? 

5.  From  the  sum  of  f  and  |  take  the  sum  of  y\  and  |. 

6.  How  many  cords  of  pine  wood  at  ^  3.25  a  cord  must 
be  given  for  12  yards  of  broadcloth  at  $  2.10  a  yard  ?  Solve 
by  cancellation. 


CANCELLATION.  345 

7.   Find  the  prime  factors  of  13860. 

13  X  16  X  42  X  51  ^  ^ 
6  X  17  X  48  X  91      * 

9.    Find  the  sum  of  the  prime  numbers  under  20. 

10.    Reduce  ^J-f-  to  lowest  terms. 

446.    1.    Write  in  Roman  notation  1894. 

2.  Write  the  prime  numbers  from  1  to  18  inclusive. 

3.  If  3  boxes  of  oranges  cost  $  5f ,  how  many  boxes  can 
be  bought  for  f  17  ? 

4.  A  farmer  sold  64  sheep,  and  had  f  of  his  flock  left. 
How  many  had  he  left  ? 


6.  How  many  barrels  of  flour  at  $  6  a  barrel  must  be 
given  for  3  pieces  of  linen,  each  containing  36  yd.,  at  25  ct. 
a  yard  ? 

7.  A  farmer  sold  at  market  15  sheep  at  $  y  each,  and 
bought  7  yards  of  cloth  at  $  1-|  per  yard.  How  much 
money  did  he  have  left  ? 

8.  From  the  sum  of  5^,  9f,  11|,  take  the  difference 
between  32  and  13f. 

9.  Reduce  to  their  least  common  denominator  ^J,  U, 

21      24 
"5^"8'  ■96- 

10.    Write   a   receipt   for   $20   paid  you  by  Mr.   John 
Dixon. 

447.    1.    Add  $49.50,  $43.62^,  $75.05,  $64.75,  $35.09, 
$6,031  $42,  $73.98,  $105.60. 

2.  How  many  sheep  at  $5  each  must  be  given  for  15 
horses  at  $  150  each  ? 


346  TOPICAL   REVIEW. 

3.  What  is  tlie  sum  of  ^  +  ^  +  |? 

4.  From  f  of  ^  of  3  take  f  of  If 

6.  At  $  39 J  apiece,  liow  many  cows  can  be  bought  for 
^2504^? 

6.  How  many  times  is  -^-^  of  f  of  6^  contained  in  |  of 
54xt-*-|? 

7.  Define  multiple  and  greatest  common  divisor. 

8.  Give  and  define  proper  fraction.     Mixed  number. 

9.  If  a  man  spends  |  of  his  money  for  a  house,  ^  for  a 
farm,  and  has  $  3400  in  cash  left,  what  is  the  amount  of  his 
wealth  ? 

10.  A  grocer  bought  3  barrels  of  apples  of  different 
qualities  at  $  2.75,  $  3.12,  and  $  3.25  a  barrel.  What  was 
the  average  cost  ? 

448.    1.    What  is  reduction  of  fractions  ? 

2.  Eeduce  {^  to  156ths. 

3.  Express  -^-^-f-^  in  its  simplest  form. 

4.  Change  to  fractions  having  the  least  common  denomi- 
nator, 7,  _9^,  and  j\. 

5.  If  a  merchant  buys  tea  at  $  f  a  pound,  and  sells  it  at 
$  f ,  does  he  gain  or  lose,  and  how  much  ? 

6.  Find  the  sum  of  f,  7f  and  8f . 

8.  A  man  engaged  to  labor  30  days,  but  was  absent  5^ 
days.     How  many  days  did  he  work  ? 

9.  A  young  man  received  a  salary  of  $  60f  a  month, 
and  paid  for  his  board  $  30^,  for  washing  $  1  J,  and  for 
other  expenses  $  i^j^^.     How  many  dollars  does  he  save  ? 


FRACTIONS.  34T 

449.  1.   Define  a  proper  fraction,  and  give  an  example  of 
one. 

2.  A  merchant  bought  three  pieces  of  cloth  containing 
125J,  96|-,  and  48|  yards.     How  many  yards  did  he  buy  ? 

3.  What  is  the  value  of  2i  times  f  of  |  of  1^  ? 

4.  If  9  men  consume  f  of  9|  pounds  of  meat  in  a  day, 
how  much  does  one  man  consume  ? 

5.  A  farmer  distributed  15  bushels  of  corn  among  sev- 
eral persons,  giving  them  1|-  bushels  apiece.  Among 'how 
many  persons  did  he  divide  it? 

113 

6.  What  is  the  value  of  — r-^  ? 

4 

T 

7.  What  number  must  be  added  to  22|  that  the  sum 
may  be  99^  ? 

8.  A  can  do  a  piece  of  work  in  8  days,  and  B  can  do  it 
in  6  days.     In  what  time  can  they  do  it  working  together  ? 

9.  A  pole  stands  ^  in  the  mud,  J  in  the  water,  and  21 
feet  above  the  water.     What  is  its  length  ? 

10.    A  man  bequeathed  to  his  son  $  3500,  which  was  f  of 
what  he  left  his  wife.     How  much  did  he  leave  his  wife  ? 

450.  1.    If  f  of  a  farm  is  valued  at  $1728,  what  is  the 
value  of  the  whole  ? 

2.  If  8  be  added  to  both  terms  of  the  fraction  f,  will  its 
value  be  increased  or  diminished,  and  how  much  ? 

3.  If  the  sum  of  two  fractions  is  |,  and  one  of  them  is 
^,  what  is  the  other  ? 

4.  Express  in  its  simplest  form  the  quotient  of  2025 
divided  by  3645. 

5.  If  the  dividend  is  J  and  the  quotient  ^-^,  what  is  the 
divisor  ? 


348  TOPICAL  REVIEW. 

6.  At  $  J  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $51?     (Analysis.) 

7.  Define  Fraction,  Terms  of  a  Fraction,  Improper  Frac- 
tion, Compound  Fraction,  and  Complex  Fraction. 

8.  Change  ^-^  to  a  whole  or  mixed  number. 

9.  How  many  8ths  of  a  bushel  in  9^  bushels  ? 

10.  Change  |,  i,  -^-^,  -f^,  ^  to  equivalent  fractions  having 
a  common  denominator. 

451.  1.  A  farmer  sells  6  jars  of  butter  each  holding  8 
pounds,  at  36  j^  a  pound,  and  receives  in  payment  14  cans 
of  coffee,  each  holding  two  pounds.  What  was  the  price  of 
the  coffee  ?     Work  by  using  cancellation. 

2.  If  a  man  walks  3^  miles  in  one  hour,  how  far  can  he 
walk  in  9  hours  ? 

3.  Find  the  sum  of  13|,  \\,  6|,  20ij,  and  /g. 

4.  How  many  days'  work  at  $1|  a  day  will  pay  for  8i|- 
yards  of  cloth  at  $21  a  yard,  and  5^  lb.  of  butter  at  25 
cents  a  pound? 

5.  The  product  of  two  numbers  is  41|,  and  one  of  them 
is  160yf4.     What  is  the  other  ? 

6.  Find  the  sum  of  93567  +  20754867  +  4756  +  925674 
+  6543987  +  6579  +  98675  +  567923  +  645876  +  9346  + 
878  +  54562  +  888. 

7.  Tell  in  words  what  these  numbers  are:  1950;  90; 
4040;  73000007. 

8.  Find  the  difference  between  76392  x  4506  and  985301 
X976. 

9.  What  will  79  ten-ton  cars  of  coal  be  worth  at  $5.50 
a  ton? 

10.  If  you  should  buy  376  horses  for  $  65123,  how  much 
would  you  sell  them  for  apiece  to  gain  $  5189  ? 


DECIMALS.  349 

DECIMALS. 

452.  1.  A  merchant  bought  four  pieces  of  cloth  contain- 
ing 32|,  38|-,  40|-,  45f  yards,  respectively.  How  many 
yards  did  he  buy? 

2.  Change  to  decimals  and  add :  J,  j,  4|. 

3.  From  a  farm  containing  128|-  acres,  84|-  acres  were 
sold.  How  many  were  left  ? 

4.  Find  the  prime  factors  of  1008. 

5.  50h-.05  =  ? 

6.  What  will  2.47  pounds  of  coffee  cost  at  f  .48  per 
pound  ? 

7.  If  one  yard  of  ribbon  costs  34|-  cents,  what  will  6 
pieces  cost,  each  piece  containing  13.12  yards  ? 

8.  Write  in  words  68.0642. 

9.  If  9  yards  of  cloth  cost  $1.17,  what  will  15  yards 
cost? 

10.  A  lady  went  shopping  with  $45.  She  paid  $4J^ 
for  shoes,  $5|-  for  a  hat,  $12J  for  a  dress.  How  much 
money  had  she  left  ? 

453.  1.  If  a  farm  is  worth  $3200,  how  much  is  f  of  it 
worth  ? 

2.  From  one  million  take  one  millionth. 

3.  What  is  the  difference  in  cents  between  f  of  a  dollar 
and  f  of  a  dollar  ? 

4.  Point  off  into  periods  96308796,  and  write  over  each 
period  its  name. 

5.  Express  with  figures  the  following  numbers:  Seven 
million  ninety-five  thousand,  sixty-three  and  fifteen  thou- 
sandths, and  seven  hundred  and  seven  hundredths. 

6.  Read,  and  write  in  words,  the  following :  642.0016  j 
100.01. 


350  TOPICAL   REVIEW. 

7.  353812416 -V- 589  =  ? 

8.  Find  the  sum  of  684.8,  96.84,  6.075,  .1906,  7508. 

9.  At  $9  per  M.,  what  will  6728  feet  of  lumber  cost  ? 
10.  At  $.65  per  C,  what  will  1240  pens  cost  ? 

■I  of  4 
454.   1.   Reduce  to  a  simple  fraction    ^      ^   - 


2.  What  fraction  of  llf  is  5|  ? 

3.  What  common  fraction  equals  .0125  ? 

4.  Reduce  ^y\  to  a  decimal. 

5.  A  farmer  sold  120  sheep,  which  were  |-  of  his  flock. 
How  many  had  he  before  the  sale  ? 

6.  A  grocer  sold  J  of  a  barrel  of  sugar  to  one  man  and 
J  of  it  to  another,  and  had  80  pounds  left.  How  many 
pounds  did  the  barrel  contain  at  first  ? 

7.  A  and  B  can  do  a  piece  of  work  in  12  days;  A  can  do 
it  in  25  days.     In  how  many  days  can  B  do  it  ? 

8.  A  man  spent  f  of  his  money  for  a  horse  and  |  of  the 
remainder  for  a  buggy  and  harness,  and  had  $37.50  left. 
How  much  money  had  he  at  first  ? 

9.  In  dividing  by  a  decimal,  how  do  you  determine  the 
proper  place  of  the  decimal  point  in  the  quotient  ? 

10.    At  $9.75  per  thousand,  what  will  16544  bricks  cost  ? 

455.    1.    Which   is  the  greater,  if  ^^  ft  ^     How  much 
greater  ? 

2.  Find  the  sum  of  78|,  87^^,  4|,  and  79f 

3.  A  man  has  three  lots,  which  are  120,  420,  and  600  ft. 
wide  respectively.  He  wishes  to  divide  them  into  lots  of 
the  greatest  equal  width  possible.  How  wide  will  each  lot 
be  ?     How  many  such  lots  can  he  make  ? 


DECIMALS.  351 

4.  If  .375  of  a  ton  of  coal  cost  $  2.40,  what  is  the  price 
of  one  ton  ?     How  many  tons  can  be  bought  for  $80  ? 

5.  Reduce  to  decimals  f,  ^,  |,  y-,  -^^. 

6.  Write  in  figures  thirteen  thousandths,  four  hundred 
and  five  hundredths,  five  hundred  fifteen  millionths,  and 
add  the  results. 

7.  Eeduce  to  a  simple  fraction  ^ ^. 

8.  Sold  a  house  for  $4797,  which  was  two-sevenths 
more  than  it  cost.     Find  the  cost  price. 

9.  Make  a  bill  for  the  following  articles,  bought  to-day 
of  James  Brown,  No.  23  Warburton  Avenue,  Yonkers, 
N. Y. :  30  oranges  at  25  cents  a  dozen ;  7  lb.  of  coffee  at 
28  ^ ;  3^  lb.  prunes  at  13  ^ ;  1  bag  of  sugar  containing  28  lb. 
at  51  ^.  Receipt  the  bill  as  though  you  were  James  Brown's 
clerk.  » 

10.  Bought  three  boxes  of  oranges  containing  263,  220, 
and  156  oranges,  respectively,  at  $3.50  per  hundred,  and 
sold  them  for  50  ^  per  doz.     Find  the  amount  of  profit. 

(7f-2.05)^(5x.23) 
**'°'    ■"•      .7|  +  2.23^  -  .6 -^  .4    * 

2.  1000-^.001  =  ? 

3.  1.  + J +  .75  4- If +  .330=? 

4.  Reduce  -f^  to  a  decimal. 

5.  A  +  Tl(7-TU7  +  Y'Tr  +  TV  =  ? 

6.  ^  X  10.0019  X  1.2  X  f  X  .463  =  ? 

7.  Change  .0507  to  hundredths. 

8.  Change  8.84  to  a  common  fraction  in  its  lowest 
terms. 

9.  .123  -  .01  -  .11  -  .003  =  ? 
10.   .0509 -f  ^-.27  =  ? 


352  TOPICAL   REVIEW. 


457.  1.    (.05015  ^  2.006)  +   (24.6  -   .0012   x   yV)   - 
1200f  =  ? 

2.  Express   in  words   the  following:    10020.00042024; 
.000702;  .00000018;  30000.00030;  .00010020. 

3.  If  a  man  travels  at  the  rate  of  7.4  miles  an  hour, 
how  long  will  it  require  to  travel  370  miles  ? 

4.  What  will  be  the  cost  of  3|-  yd.  of  cloth  at  .75  dollars 
per  yard  ? 

5.  Find  the  sum  of  .125,  46.42,  9.3,  164.25,  .80406. 

6.  From  1000  subtract  .001. 

7.  Find  the  cost  of  445.375  bushels  of  wheat  at  $.9173 
per  bushel. 

8.  Change  .00125  to  a  common  fraction. 

9.  Eeduce  ^-^  to  a  decimal. 

10.    At  $  .044  per  pound,  how  many  pounds  of  sugar  can 
be  bought  for  $44? 

458.  1.    Find  the  cost  of  9^  tons  of  coal,  if  J  of  a  ton 
cost  $3.00. 

2.  Find  the  sum  of  40  units,  20  tens,  464  thousandths, 
5  ten-thousandths,  and  1  millionth. 

3.  Write   in   figures   two   and   twenty-six  hundredths; 
two  and  twenty  six-hundredths. 

4.  What  number  multiplied  by  14^  will  produce  1684|  ? 

5.  If  f  of  a  yacht  is  valued  at  $3840^,  what  is  the  value 
of  the  whole  ? 

6.  If  I  of  a  pound  of  tea  cost  $.50,  what  will  16| 
pounds  cost  ? 

7.  Reduce  to  simplest  form : 

*       6J  ^  llf 


DENOMINATE   NUMBERS.  353 

8.  Eeduce  ||f  to  a  decimal. 

9.  A  man  bequeathed  -^-^  of  his  estate  to  his  elder  son, 
and  the  remainder  to  his  younger  son,  who  received  $  1344. 
What  was  the  estate  worth  ? 

10.    What  must  be  paid  for  8960  pounds  of   plaster   at 
$5.50  per  ton? 

DENOMINATE    NUMBERS. 

459.  1.    Define  simple  quantity  ;  compound  quantity. 

2.  Reduce  1760  cwt.  to  higher  denominations. 

3.  Add:  11  oz.  11  pwt.  15  gr. ;  7  oz.  12  pwt.  19  gr. ; 
10  oz.  13  pwt.  17  gr. 

4.  Write  the  table  of  long  measure. 

5.  Find  the  total  area  in  sq.  yards  of  the  ceiling  of  a 
room  18  ft.  long  and  15  ft.  wide. 

6.  Find  the  number  of  square  feet  in  the  surface  of  a 
cube  3  ft.  by  3  ft.  by  3  ft. 

7.  Find  the  total  area  in  the  four  walls  of  a  room  18  ft. 
long,  15  ft.  wide,  and  9  ft.  high. 

8.  Define  fraction ;  mixed  number ;  proper  fractions ; 
improper  fractions. 

9.  Reduce  f,  f,  and  /^  to  similar  fractions. 
10.    Define  circumference  ;  diameter. 

460.  1.    Wliat  is  the  value  of  J  of  }  divided  by  i  of  | 
plus  I  of  I  ? 

2.  A  and  B  can  build  a  shop,  working  together,  in  10 
days ;  B  can  build  it,  working  alone,  in  30  days.  In  how 
many  days  can  A  build  it  ? 

3.  Add  0.525  mi.,  0.125  rd.,  0.5  yd.,  and  0.16  ft. 

4.  From  ^^  of  a  square  rod  take  |-  of  a  square  yard. 


354  TOPICAL   REVIEW. 

5.  Find  ^  of  9- A.  70  sq.  rd.  15  sq.  yd.  7  sq.  ft.  19  sq.  in. 

6.  There  is  a  room  15  ft.  long,  12  ft.  wide,  and  9  ft. 
high.  It  has  2  windows,  each  3  ft.  by  6  ft.,  and  a  door  3  ft. 
by  7  ft.  Taking  out  the  space  for  door  and  windows,  how 
much  will  it  cost  to  plaster  this  room  at  25/  per  square 
yard? 

What  will  be  the  cost  of  floor  boards  1^  in.  thick,  to  lay 
the  floor  of  this  room  at  $  40  per  thousand  ? 

7.  There  is  a  square  field  40  chains  around.  How  many 
acres  are  in  it  ? 

8.  In  a  space  27  ft.  long,  18  ft.  wide,  and  12  ft.  high, 
there  may  be  placed  how  many  cubes  3  feet  on  each  edge  ? 

9.  How  many  grains  in  a  ton?  How  many  gallons  in 
a  cu.  yard  ? 

10.    How  many  grains  in  a  Troy  pound  ? 

461.    1.    What  decimal  of  a  mile  is  ^  of  5  mi.  89  rd.  3  yd. 
2ft.? 

2.  Divide  15  T.  17  cwt.  29  lb.  7  oz.  by  f 

3.  36^  sq.  in.  equals  what  fraction  of  an  acre? 

4.  i  mi.  +  f  rd.  +  i  ft.  -  7i  yd.  =  ? 

5.  If  7  spoons  weigh  7  oz.  12  pwt.  9  gr.,  what  will  13 
similar  spoons  weigh  ? 

6.  Add  36|,  .00125, 1460,  f ,  j\,  and  16.26. 

7.  If  I  of  a  ship  is  worth  $  6285,  what  is  -^  worth  ? 

8.  To-day  you,  as  a  clerk  of  Chester  &  Wilson,  sell 
Wm.  Lambert  20  bbl.  flour  at  $4.87^-,  4500  lb.  meal  at 
$  1.06,  per  cwt.,  and  2450  lb.  bran  at  $  13.50  per  T.  Make 
out  the  proper  bill. 

9.  Define  improper  fraction;  decimals;  reduction  de- 
scending ;  a  biU. 

10.   Keduce  £  17  14  s.  Sfar.  to  farthings,  and  prove. 


DENOMINATE   NUMBERS.  355 

462.  1.  For  22  lb.  14  oz.  of  butter  worth  16^  a  pound 
a  man  gets  12  quarts  of  syrup.  -  What  is  the  price  of  the 
syrup  per  gallon  ? 

2.  At  $3  a  perch,  what  will  masons  earn  in  laying  a 
wall  8  ft.  high  and  2  ft.  thick  in  a  cellar  dug  36  ft.  x 
42  ft.  ? 

3.  At  60^  per  yd.,  what  will  be  the  least  cost  to  carpet 
a  room  14  ft.  x  16  ft.  with  ingrain  carpet,  using  only  full 
breadths,  and  no  waste  for  cutting  ? 

4.  Eeduce  .875  of  a  bushel  to  lower  denominations. 

5.  How  many  bushels  will  a  bin  contain  that  is  9  ft. 
long,  4  ft.  wide,  and  6  ft.  deep  ? 

6.  How  much  will  a  piece  of  land  20  rd.  by  18  rd.  cost 
at  $  116  per  A.  ? 

7.  Find  the  cost  of  a  Brussels  carpet  (27  in.  wide)  at 
$  1.15  per  yd.  for  a  room  16  ft.  by  23  ft.,  breadths  to  run 
crosswise. 

8.  At  f  .60  per  sq.  yard,  what  will  it  cost  to  plaster 
the  sides  and  ceiling  of  a  room  18  x  12  x  8  ft.  ? 

9.  Leaving  Dubuque  I  travel  until  my  watch  is  1  h.  20 
min.  slow.     Which  way,  and  how  far,  have  I  travelled  ? 

10.  My  cistern  is  8  ft.  by  4-1-  ft.  When  the  water  is  27 
in.  deep,  how  many  barrels  of  water  is  there  in  the  cistern  ? 

463.  1.    Define  and  illustrate  decimal ;  multiple ;  quotient, 

2.  If  I  burn  a  pint  of  kerosene  every  night,  what  will 
a  three  weeks'  supply  cost  me  at  15  cents  a  gallon  ? 

3.  Find  the  sum  of  J  mi.  i  rd.  f  ft. 

4.  How  many  boards,  each  15  feet  long,  will  be  required 
to  build  56^  rods  of  fence  four  boards  high  ?     Analyze. 


356  TOPICAL   REVIEW. 

5.  Find  the  value  of  f  of  a  chest  of  tea  weighing  57^ 
pounds,  at  $  1^  per  pound.  > 

^     ^  ,       14  X  32  X  96  X  7  X  163      „ 
^'    ^^^"^192x21x28x55x8  =  - 

7.  How  many  times  will  a  wheel  12  ft.  4  in.  in  circum- 
ference revolve  in  going  10  miles  ? 

8.  How  many  days  must  a  laborer  work,  at  $  1.12^  a 
day,  to  pay  for  6  cords  of  wood,  at  $  3.37|^  per  cord  ? 

9.  A  man  was  born  Feb.  29, 1844,  and  died  Mar.  15, 1880. 
How  many  birthdays  did  he  have  ?     What  was  his  age  ? 

10.  What  is  the  product  of  12  millionths  multiplied  by 
12  thousandths  ? 

464.  1.  How  many  pickets  3  in.  wide,  placed  3  in.  apart, 
will  be  required  for  a  fence  around  a  rectangular  yard  4  rd. 
6  ft.  long,  and  3  rd.  8  ft.  wide  ? 

2.  A  farmer  has  a  piece  of  land  containing  7|f  acres, 
fenced  in  the  form  of  a  rectangle,  its  length  being  twice  its 
width.     What  are  the  dimensions  of  the  rectangle  ? 

3.  Oswego  County  has  an  area  of  970  square  miles, 
and  a  population  of  71780.  What  is  the  population  to  the 
square  mile  ?  How  many  acres  could  be  given  to  each  one 
of  the  entire  population  ? 

4.  Oswego  is  in  longitude  76°  35'  W.,  Albany,  73°  32' 
W.  What  is  the  difference  in  their  longitude  ?  When  it 
is  noon  by  the  sun  in  Albany,  what  o'clock  is  it  in  Oswego  ? 

5.  What  will  be  the  cost  of  carpeting  a  room  18  ft.  long 
and  12  ft.  wide  with  Brussels  carpet  f  yd.  wide,  at  85^  a 
yd.,  the  strips  to  run  lengthwise  of  the  room,  and  allowing 
4  in.  to  be  turned  under? 

6.  At  $  25  per  thousand,  what  is  the  value  of  16  planks, 
each  18  ft.  long,  6  in.  wide,  2^  in.  thick  ? 


DENOMINATE   NUMBERS.  357 

7.  rind  the  cost  of  5  pieces  of  timber,  each  48  ft.  long, 
9  in.  by  12  in.,  at  $  1.50  per  hundred  board  ft. 

8.  How  many  board  feet  of  lumber  will  be  required  to 
fence  a  lot  80  ft.  by  40,  the  boards  being  10  ft.  by  6  in.,  and 
the  fence  4  boards  high  ? 

9.  How  many  board  feet  will  it  take  to  cover  the  top  of 
a  tank  14  ft.  long,  6  ft.  wide,  with.planks  2  in.  thick  ? 

10.  A  man  sold  two  bushels  of  strawberries  as  follows : 
to  Mrs.  A.  he  sold  -^^  of  the  berries,  to  Mrs.  B.  f ,  and  the 
remainder  to  Mrs.  C.     How  many  quarts  did  Mrs.  C.  buy  ? 

465.  1.  Two  telegraph  stations  are  18  miles,  224  rods 
apart.  If  the  telegraph  poles  between  the  stations  are  8 
rods  apart,  how  many  poles  will  be  needed,  and  how  much 
will  they  cost  at  50^  apiece  ? 

2.  What  is  the  value  of  a  triangular  piece  of  land, 
having  a  base  of  60  chains  and  an  altitude  of  40  chains,  at 
$  60  per  acre  ? 

3.  How  many  times  can  a  dish  holding  2  qt.  i  pt.  be 
filled  from  a  jar  holding  3  gal.  2  qt.  1  pt.  ?  How  much 
will  be  left  in  the  jar  ? 

4.  Find  the  cost  of  carpeting  a  room  24  ft.  long  and 
18  ft.  wide,  with  carpet  27  inches  wide,  the  strips  running 
lengthwise  of  the  room,  the  cost  of  the  carpet  being  $  1.65 
a  yard,  and  no  loss  in  matching  the  figures. 

5.  After  spending  f  46|,  I  had  |  of  my  money  left. 
How  much  had  I  at  first? 

6.  A  man  traded  7  wagons  at  $  71^  each  for  84  bbl.  of 
flour.     What  was  the  flour  per  barrel  ? 

7.  What  is  the  capacity  in  liters  of  a  cistern  1.5  meters 
long,  9  decimeters  wide,  and  86  centimeters  deep  ? 


358  TOPICAL  REVIEW. 

8.  How  many  bricks  8  in.  long,  4  in.  wide,  and  2  in. 
thick  will  it  take  to  pave  a  section  of  street  200  ft.  long, 
36  ft.  wide,  the  bricks  being  placed  on  their  edges  ? 

How  much  will  the  bricks  cost  at  $  7.35  per  M.  ? 

9.  How  many  cubic  inches  in  a  bin  which  contains  300 
bu.  of  wheat  ? 

10.  The  distance  around  a  circular  park  is  1|-  miles. 
How  many  acres  does  it'  contain? 

466.  1.  How  many  blocks  i|-  of  a  foot  long  can  be  cut 
from  a  board  22  ft.  long  ? 

2.  How  many  poor  families  can  be  supplied  with  ^  of 
a  ton  of  coal  each  from  12  tons  ? 

3.  How  many  pairs  of  tray-cloths,  each  containing  }  of 
a  yard,  can  be  cut  from  15  yards  of  linen  ? 

4.  In  how  many  months,  paying  $J  per  week,  will  a 
debt  of  $  36  be  paid  ? 

5.  J  is  what  part  of  |  ? 

6.  A  37-gallon  cask  is  f  full ;  6^  gallons  being  drawn 
off,  how  full  will  it  be  ? 

7.  If  from  a  piece  of  cloth  containing  96  yd.  you  sell 
24f  yd.,  what  fractional  part  of  the  piece  remains  ? 

8.  llf  bushels  are  what  fraction  of  15|  bushels  ? 

9.  fiis  what  part  of  I  of  J? 

10.  A  man  had  700  head  of  cattle.  He  sold  at  one  time 
50  head,  at  another  75  head.  What  fraction  of  the  whole 
did  he  sell  ? 

467,  1.  How  many  cubic  feet  of  stone  will  it  take  to 
build  the  walls  of  a  cellar  36  ft.  long,  24  ft.  wide,  and  8 
ft.  high,  outside  measurement,  the  walls  being  18  in.  thick  ? 
How  much  will  the  stone  cost  at  $  4.50  per  cord  ? 


DENOMINATE   NUMBERS.  359 

2.  Find  the  diameter  of  a  wheel  whose  circumference  is 
50  feet. 

3.  If  1  bu.  3  pk.  6  qt.  of  walnuts  cost  $  3.10,  what  is  the 
price  per  quart  ? 

4.  What  will  be  the  cost  of  5  gal.  3  qt.  1|  pt.  of  maple 
syrup  at  75  cents  per  gallon  ? 

5.  Find  the  cost  of  5362  pounds  of  coal  at  $4.50  per 
ton. 

6.  How  long  a  time  has  elapsed  since  the  first  message 
was  sent  by  telegraph,  May  29,  1844  ? 

7.  How  much  profit  will  there  be  in  buying  4  bu.  1  pk. 
6  qt.  of  cranberries  at  $  2  a  bushel,  and  selling  them  at 
10  cents  a  quart  ? 

8.  How  many  days  will  a  6-ounce  bottle  of  medicine 
last  a  patient  who  takes  a  teaspoonful  three  times  a  day, 
a  teaspoon  holding  60  drops  or  minims  ? 

9.  Multiply  9  mi.  25  rd.  3  yd.  2  ft.  by  f . 

10.    Divide  110  mi.  149  rd.  3  yd.  2  ft.  6  in.  by  f 

468.   1.    Multiply  25  yards  2  ft.  11  in.  by  16. 

2.  From  6  bu.  6  qt.  take  3  pk.  1  qt.  1  pt. 

3.  What  is  the  difference  in  time  between  June  16, 
1890,  and  Feb.  4,  1895  ? 

4.  What  will  it  cost  to  build  the  walls  of  a  cellar  that 
is  26  ft.  long  and  16  ft.  wide,  6-|-  ft.  deep,  the  wall  being 
18  in.  thick,  at  $  1.50  a  perch  ? 

5.  A  field  is  16  ch.  10  links  long  and  5  ch.  wide.  How 
many  acres  does  it  contain  ? 

6.  How  many  board  feet  in  24  joists,  10  in.  by  2  in.  by 
16  ft.,  and  what  are  they  worth  at  $  11  per  M.  ? 

7.  What  is  a  pile  of  four-foot  wood  worth  that  is  16  ft. 
long  and  6  ft.  high,  at   $  4.50  a  cord  ? 


360  TOPICAL   REVIEW. 

8.  How  many  grains  in  5  lb.  of  butter  ? 

9.  Eeduce  12  cwt.  80  lb.  6  oz.  to  the  decimal  of  a  ton. 
10.   Find  the  sum  of  184|,  372^,  19|. 

PERCENTAGE. 

469.  1.  Express  as  %  the  following:  .28;  .065;  3.07; 
.004. 

2.  Express  decimally  the  following:  ^%  ;  6^%  ;  8%; 
125%. 

3.  From  a  farm  of  144  acres  18  acres  were  sold.  What 
per  cent  of  the  farm  was  sold  ? 

4.  A  grocer  sold  eggs  at  12|-  cents  a  dozen  and  gained 
2h(Jo.     What  was  the  cost  ? 

5.  A  man's  farm  cost  him  %  5400.  His  crop  of  potatoes 
yielded  him  in  cash  8  %  of  the  cost  of  the  farm.  What  was 
the  value  of  his  potatoes  ? 

6.  If  a  merchant  pays  $  .80  a  yard  for  a  roll  of  carpet, 
and  because  it  became  damaged  sells  it  for  $  .^^  a  yard, 
what  per  cent  does  he  lose  ? 

7.  Sent  my  agent  in  St.  Louis  $3017.60,  with  which  he 
is  to  purchase  flour  at  $4.00  per  bbl.,  after  deducting  his 
commission  at  2|-  per  cent.  How  many  barrels  should  I 
receive  ? 

8.  If  by  selling  36840  ft.  of  lumber  at  $21.12  per  M., 
you  gain  28  per  cent,  what  would  be  your  gain  or  loss  by 
selling  it  at  $17  per  M.  ? 

9.  If  a  merchant  has  marked  an  article  for  sale  at  50 
per  cent  above  cost,  what  per  cent  will  he  deduct  from  the 
asking  price  if  he  sells  the  article  at  cost  ? 

10.  $7884.00  is  to  be  raised  by  taxation  in  a  certain 
school  district.  The  taxable  property  of  the  district  is 
$584,000.  Find  the  rate  of  tax,  and  A's  tax,  whose 
property  is  assessed  at  $3850. 


PEECENTAGB.  361 

470.  1.   From  ^  of  a  week  take  ^  of  a  day. 

112 

2.  Reduce  — ^  to  a  simple  fraction. 

12f 

3.  Define  base  and  rate. 

4.  How  many  hundredths  of  anything  is  -^  of  it  ?  J  of 
it?     iofit?     i^ofit? 

5.  What  is  12%  of  1682? 

6.  Express  as  common  fractions  in  their  lowest  terms: 
25%,  62i%,  121%,  16|%. 

7.  A  speculator  bought  2160  barrels  of  apples,  and  upon 
opening  them  found  15%  of  them  spoiled.  How  many 
barrels  did  he  lose  ? 

8.  A  farmer  sold  50  sheep,  which  was  25%  of  his  whole 
flock.     How  many  sheep  had  he  at  first  ? 

9.  My  income  this  year  is  $4028,  which  is  24%  less 
than  it  was  last  year.     How  much  was  it  last  year  ? 

10.  A  commission  merchant  sells  goods  to  the  amount  of 
$  6895.     What  is  his  commission  at  3%  ? 

471.  1.  I  bought  two  houses  at  $  3500  each,  and  sold  one 
at  a  gain  of  22%,  and  the  other  at  a  loss  of  22%.  Did  I 
gain  or  lose  on  both,  and  how  much  ? 

2.  If  I  sell  for  $  16  what  cost  $  20,  what  per  cent  do  I 
lose? 

3.  If  I  buy  a  piano  for  $450,  and  sell  it  for  $  600,  what 
per  cent  do  I  gain  ?  _  ' 

4.  Define  insurance ;   premium ;  taxes. 

5.  What  will  be  the  cost  of  insuring  a  quantity  of  wheat 
valued  at  $8,450,  at  |y<,  ? 

6.  The  premium  for  insuring  a  schoolhouse,  at  the  rate 
of  1^%,  was  $  75.     For  what  sum  was  it  insured  ? 


362  TOPICAL   REVIEW. 

7.  The  town  of  B  is  to  be  taxed  $3700  to  build  a 
bridge.  The  taxable  property  is  valued  at  $1,850,000. 
What  will  be  the  rate  of  taxation,  and  the  tax  on  Mr.  A., 
whose  property  is  valued  at  $  5000  ? 

8.  What  is  the  duty,  at  25%,  on  4796  pounds  of  Eussia 
iron,  worth  10  cents  a  pound  ? 

9.  What  number  increased  by  25%  of  itself  is  506.25  ? 

10.    Find  the  net  cost  of  "a  bill  of  goods  amounting  to 
$3750  at  10%  discount,  and  4%  off  for  cash. 

472.    1.    An  agent  sold  4250  yd.  of  calico  at  3 J  ^  per  yard. 
What  was  his  commission  at  2^  %  ? 

2.  A  real  estate  broker,  who  charges  4%  commission, 
receives  $224  for  selling  a  house.  What  price  is  paid  for 
the  house  ? 

3.  If  $  8240  is  sent  to  an  agent  to  cover  the  amount  of 
his  purchase  and  his  commission  of  3%,  what  is  the  amount 
of  his  purchase  ? 

4.  An  hotel  is  insured  for  $90,000  at  2i%  for  3  years. 
What  is  the  annual  cost  of  insurance  ? 

5.  A  man's  weight  is  180  pounds,  and  he  is  20%  heavier 
than  his  brother.     What  is  his  brother's  weight  ? 

6.  A  bill  for  hardware  amounting  in  gross  to  $  2537.75 
is  subject  to  discounts  of  40%,  10%,  and  5%.  What  is  the 
net  amount  ? 

7.  If  you  remove  the  decimal  point  from  the  number 
6.45,  what  effect  does  it  produce  upon  the  number  ? 

8.  If  from  the  same  number  you  take  the  period  from 
after  the  6  and  place  it  before  the  6,  what  will  be  the  effect  ? 

9.  At  $12.75  a  ton  what  will  3265  pounds  of  hay  cost  ? 
10.    A  tree  measures  8.2  ft.  in  circumference.     What  is 

the  diameter  ? 


PERCENTAGE.  363 

473.  1.  Find  i%  of  12.00;  ^2^%  of  2000  bushels  of  corn ; 
200%  of  5  dozen  eggs ;  J  of  1  per  cent  of  100  tons  of  coal. 

2.  What  fraction  increased  by  25  per  cent  of  itself 
equals  fi-  ?       -i 

3.  What  is  the  effect  upon  the  quotient  when  both  the 
dividend  and  the  divisor  are  multiplied  by  the  same  number  ? 

4.  Express  as  fractions  in  lowest  terms,  81%,  2-^%, 
181%. 

5.  Express  as  per  cent,  using  the  sign,  .1352,  ■},  2,  3^^. 

6.  Express  as  decimals,  ^^^,  ^,  -i%,  20%,  15^%. 

7.  What  per  cent  of  the  number  of  days  in  February, 
1896,  is  the  number  of  days  in  January,  1896  ? 

8.  My  house  cost  $  6000,  which  was  400  per  cent  more 
than  I  paid  for  the  lot.     Find  the  cost  of  both. 

9.  After  spending  $  14  for  a  suit  of  clothes,  a  man  had 
$  126  left.     What  per  cent  of  his  money  did  he  spend  ? 

10.  An  agent  purchased  8^  tons  of  sugar  at  3^  cents  per 
pound  on  3%  commission.  Find  the  cost  of  the  sugar,  in- 
cluding commission. 

474.  1.  What  is  the  rate  of  taxation  on  ^1000  when 
^147000  is  raised  on  $35,000,000? 

2.  A  man  selling  cloth  at  $4.20  per  yard,  gained  20%. 
Had  he  sold  it  at  $3.60  per  yard,  would  he  have  gained  or 
lost,  and  what  per  cent  ? 

3.  If  f  of  a  mill  is  worth  $10,000,  what  is  i  of  the 
remainder  worth  ? 

4.  Bought  a  horse  for  $160 J,  and  sold  it  for  |-  of  its 
cost.     How  much  did  I  lose  ? 


364  TOPICAL   REVIEW. 

5.  Define  least  common  multiple;  improper  fraction; 
prime  factor. 

6.  Simplify?!. 

7.  Find  the  cost  of  10  sticks  of  timber,  each  16  feet 
long,  14  inches  wide,  and  10  inches  thick,  at  $  16.50  per  M., 
board  measure. 

8.  How  many  gallons  will  a  cistern  hold  that  is  12  ft. 
long,  8  ft.  wide,  and  6  ft.  deep  ? 

9.  If  9|  yards  of  cloth  are  worth  $24,375,  what  is  the 
value  of  16|^  yards  at  the  same  rate  ? 

10.    Name  the  unit  of  weight  in  the  metric  system,  and 
give  the  table  in  which  that  unit  occurs. 

475.    1.    I  spend  65  per  cent  of  my  salary,  but  am  able 
to  save  $  980.     How  much  do  I  spend  ? 

2.  How  much  must  I  send  my  agent,  that  he  may  buy 
at  1^  per  cent  commission,  400  bbl.  flour  at  $  6.75  per  bbl.  ? 

3.  Given  the  amount  and  percentage,  write  the  formula 
for  finding  each  of  the  other  terms. 

4.  What  are  like  numbers  ?     Unlike  numbers  ? 

5.  Write  an  abstract  number.  Give  the  definition  of 
abstract  number. 

6.  Write  in  words  2300406.000960. 

7.  What  kind  of  number  is  4.6  bushels  ? 

8.  A  father  divided  his  property  as  follows  :  to  his  son 
John  he  gave  ^,  to  his  daughter  Susan  ^,  to  his  wife  i,  and 
the  rest,  which  was  $  13,000,  to  endow  a  school.  What  was 
the  value  of  his  estate  ? 


PERCENTAGE.  365 

9.  I  own  a  house  that  cost  me  $3000.  It  cost  me  to 
insure  it  for  3  years  $24.  The  average  yearly  cost  of 
repairs  is  $50.  The  average  yearly  tax  is  2%  of  the  cost. 
I  can  get  5%  per  annum  for  the  $3000  invested.  The 
house  will  last  60  years.  I  receive  in  rent  for  the  house 
$300  per  annum.  If  these  conditions  are  constant,  how 
much  will  I  gain  or  lose  in  60  years  ? 

10.    A  father  is  39  years  old  and  his  daughter  13.     What 
per  cent  of  the  father's  age  is  the  daughter's  ? 

476.    1.    Write  these  per  cents  as  hundredths  :  2%,  6-J-%, 
20%,  121%. 

2.  How  many  per  cent  of  a  number  is  0.20?  0.75? 
,12i?    1.40? 

3.  What  fractions  of  a  number  (in  lowest  terms)  are 
these  per  cents  :   16|%  ?  75%  ?  331%  ?  100%  ?  and  175%  ? 

4.  Express  as  hundredths  and  as  common  fractions: 
i%  ;  f  %  ;  \%  ;  1%  ;  and  3-V%. 

5.  Trom  a  stack  of  hay  7  T.  11  cwt.  were  sold,  which 
was  75^%  of  the  whole.  How  much  did  the  stack  con- 
tain before  the  sale  ? 

6.  A  lawyer  collected  65%  of  a  debt  of  $1260,  and 
charged  5%  commission  on  the  sum  collected.  What  did 
the  creditor  receive  ? 

7.  If  a  hat  that  cost  $5  be  sold  for  $9,  what  is  the 
gain  per  cent  ? 

8.  How  many  days  from  Sept.  16,  1892,  to  Feb.  12, 
1894  ? 

9.  874  is  33  J  %  less  than  what  number  ? 

10.    Eequired  the  cu.  feet  of  a  box  6  ft.  6  in.  by  4  ft.  9  in. 
by  3  ft.  3  in. 


366  TOPICAL  BEVIEW. 

477.  1.  "Write  the  following  numbers  and  add:  six  thou- 
sand sixteen  and  sixty-five  thousandths,  four  hundred  one 
thousand  forty-one  and  one-tenth,  six  hundred  one  and  nine 
hundredths,  ten  thousand  one  hundred  seventeen  and  nine 
hundred  three  thousandths,  forty-nine  hundred  forty-nine 
and  nine-tenths. 

2.  Write  in  words  83.4937007^^,  1001001.01,  90019^^. 

3.  Find  the  number  of  which  160  is  f. 

4.  Find  the  exact  number  of  days  from  July  4,  1893,  to 
to-day. 

5.  Multiply  7  lb.  8  oz.  15  pwt.  by  15. 

g  -.      18  X  963  X  44  X  27  X  2800  ^  ^ 
63  X  88  X  105  X  1926  x  45      " 

7.  Define  commission,  also  brokerage;  and  state  on 
what  sum,  or  value,  both  are  computed. 

8.  Express  decimally  274  and  y^.  Find  their  product 
as  decimals,  and  as  common  fractions,  expressing  both 
answers  decimally. 

9.  Fruit  was  sold  at  12|-)^  per  quart,  which  was  200  per 
cent  of  its  cost.  What  was  the  cost  per  bushel,  and  what 
was  the  rate  per  cent  of  profit  ? 

10.  An  agent  sold  840  bu.  of  grain  at  60^  per  bushel. 
His  commission  was  $15.12.     Find  the  rate  of  commission. 

478.  1.  A  man  owes  you  a  debt  of  $2160,  which  he  de- 
clines to  pay.  Your  lawyer  succeeds  in  collecting  70  per 
cent  of  the  debt,  and  charges  5  per  cent  commission  for 
his  services.     What  sum  do  you  receive  ? 

2.  A  manufacturer  sent  $  1287.50  to  a  commission  mer- 
chant who  charges  3  per  cent  commission,  instructing  him 
to  purchase  wool  at  $  0.33^  per  pound.  How  many  pounds 
of  wool  will  be  received  ? 


PERCENTAGE.  367 

3.  A  farm  was  sold  for  $  8000,  which  was  20  per  cent 
less  than  its  real  value.  If  it  had  sold  at  $  12000,  what 
per  cent  above  its  real  value  would  it  have  brought  ? 

4.  A  commission  merchant  sold  for  a  farmer  6000  lb.  of 
pork  at  S^^  per  pound.  He  charged  1^%  commission  for 
selling,  and  paid  $18.81  for  freight.  How  many  feet  of 
pine  boards  at  $  25  per  1000  ft.  could  he  purchase  with  the 
proceeds  of  the  pork,  after  deducting  1  per  cent  commis- 
sion for  buying  ? 

5.  Keduce  to  simple  fraction  in  lowest  terms : 


X 


6 


6.  What  per  cent  of  3  is  |  ?     Of  f  is  f  ?     Of  80  is  50  ? 

7.  A  drover  sold  250  sheep  for  $1150,  which  was  15% 
more  than  they  cost.  What  was  the  cost  per  head  of  the 
sheep  ? 

8.  If  20%  be  lost  on  a  ton  of  rye  straw  sold  for  $  19.20, 
what  is  the  cost  of  the  straw  per  ton  ? 

9.  How  many  per  cent  of  a  number  is  0.15  ?  0.06^  ? 
0.50?    2.25? 

10.  What  common  fraction  of  a  number  in  its  lowest 
terms  is  20%  ?  50%  ?  6i%  ?  66|%  ?  160%  ? 

479.  1.  A  man  sold  $8400  worth  of  merchandise,  and 
had  30%  of  his  stock  left.  What  was  his  entire  stock 
worth  ? 

2.  A  nferchant  sold  goods  at  20%  and  5%  off,  and  still 
made  20%  on  the  cost.  What  was  the  cost  price  of  a  book 
that  was  marked  $  1.00  ? 

3.  Bought  1000  pounds  of  butter  at  18  ^,  and  sent  it  to 
an  agent  who  sold  it  at  21)^  on  a  5%  commission.  What 
was  my  rate  of  gain  ? 


368  TOPICAL  REVIEW. 

4.  Mr.  Brown  has  a  flock  of  940  sheep  in  three  fields. 
In  the  first  are  20%  of  the  entire  flock,  in  the  second  40%, 
and  the  remainder  in  the  third.  How  many  sheep  are 
there  in  each  field? 

5.  A  lady  has  a  salary  of  ^  825  a  year.  She  spends  20% 
of  it  for  board,  35%  of  it  for  other  expenses,  and  saves  the 
remainder.     What  sum  does  she  save  ? 

6.  What  per  cent  of  a  leap  year  is  the  time  from  Wash- 
ington's Birthday  to  the  Fourth  of  July  ? 

7.  The  Barber  Asphalt  Company  engaged  to  pave  a 
street  5  miles  long  at  $55,000  a  mile.  If  the  actual  cost 
is  $  130  per  rod,  what  is  the  gain  per  cent  ? 

8.  A  commission  merchant  charges  1^%  for  selling, 
and  2}%  for  guaranteeing  the  payment  of  the  money.  His 
commission  on  a  certain  transaction  amounted  to  $384.75. 
Required  the  amount  of  the  sale. 

9.  I  bought  1100  tons  of  coal  at  $3J  per  ton.  I  sold 
40%  of  it  at  a  gain  of  50%,  40%  of  the  remainder  at  a 
gain  of  35%,  and  lost  10%  on  the  rest.  What  was  my 
actual  gain  ? 

10.    An  article  bought  at  18%  below  the  asking  price  is 
sold  for  the  asking  price.     What  is  the  gain  per  cent  ? 

INTEREST    AND    DISCOUNT. 

480.     1.    Find  the  amount  of  $975  for  1  year,  4  months, 
and  12  days,  at  6  per  cent  interest. 

2.  Find  the  interest  on  $128.45  from  March  2,1895, 
to  Dec.  14,  1895,  at  6  per  cent. 

3.  A  pile  of  wood  256  feet  long,  4  feet  wide,  and  5  feet 
high  is  sold  for  $160.     What  is  the  price  per  cord  ? 

4.  Define  per  cent ;  interest ;  proper  fraction. 

5.  State  the  difference  between  a  prime  and  a  composite 
number. 


INTEREST   AND   DISCOUNT.  869 

'6.  Find  the  cost  of  6  gal.  3  qt.  and  1  pt.  of  syrup  at 
46  cents  per  gallon. 

7.  1521  is  how  many  times  13  ? 

8.  What  is  the  interest  on  $1200  for  2  yr.  3  mo.  18  da. 
at6%?     The  amount? 

9.  What  is  the  interest  on  $1240  from  March  3  to  Aug. 
28,  at  6%  ? 

10.  Write  the  United  States  rule  for  computing  the 
amount  due  on  a  note  when  partial  payments  have  been 
made. 

481.  1.  In  what  time  will  $3960  earn  $770  at  5%, 
simple  interest  ? 

2.  If  $675,  at  simple  interest,  gain  $172.80  in  3  years, 
2  months,  12  days,  what  is  the  rate  of  interest  ? 

3.  When  interest,  time,  and  rate  are  given,  how  may  the 
principal  be  found  ? 

4.  Define  the  present  worth  and  true  discount  of  a 
debt.  Define  compound  interest,  and  make  and  solve  an 
example  to  illustrate  your  definition. 

5.  A  merchant  sells  goods  amounting  to  $6784.00  on  a 
year's  credit.  If  money  is  worth  8%,  what  sum  should  he 
accept  in  payment  of  the  bill  6  months  before  it  becomes 
due? 

6.  Write  a  negotiable  promissory  note  signed  by  James 
Fox  for  $  875.60  due  90  days  from  June  19,  payable  to  your- 
self, at  a  bank.  Name  (a)  the  payed  ;  (6)  the  drawer ;  (c)  the 
date  when  the  note  matures  (becomes  due).  What  words  in 
the  note  make  it  negotiable  ?     What  does  negotiable  mean  ? 

7.  If  you  should  sell  the  note  (Ex.  6)  to  Mr.  F.  P. 
Weaver,  what  indorsement  must  you  write  upon  it? 
Where  should  indorsements  be  written  ? 


370  TOPICAL  REVIEW. 

8.  If  the  note  is  not  paid  until  Sept.  15,  1895,  how 
much  interest  will  then  be  due  on  it? 

9.  A  farmer  expended  $5460  in  improvements  on  his 
farm,  which  was  24^  more  than  f  of  the  cost  of  the  farm. 
Find  the  cost  of  the  farm. 

10.  Principal,  interest,  and  time  being  given,  how  is  the 
rate  found  ? 

482.  1.  Find  the  amount  of  $496.85  for  2  years,  4 
months,  and  15  days  at  4  per  cent. 

2.  How  long  will  it  take  $750  at  6  per  cent  to  gain 
$67.50  interest? 

3.  A  dealer  bought  65  lawn-mowers  at  $4.25  each,  and 
sold  them  at  $3.87J  each.     What  per  cent  did  he  lose  ? 

4.  If  a  cellar  is  38  ft.  long  and  28  ft.  wide  inside  the 
wall,  and  the  wall  is  8  ft.  high  and  18  in.  thick,  how  many 
cubic  yards  of  masonry  does  the  wall  contain  ? 

5.  What  per  cent  of  a  number  equals  f  of  the  number  ? 
What  part  of  a  number  equals  33|^  per  cent  of  it  ? 

6.  Write  decimally,  6%  ;  one  hundred  six  per  cent. 

7.  A  town  6  miles  long  and  4i  miles  wide  is  eqaal  to 
how  many  farms  of  80  acres  each  ? 

8.  What  number  must  be  subtracted  from  four  hundred 
sixty-seven  thousand  six  hundred  thirty-three  to  make  it 
exactly  divisible  by  758  ? 

9.  Find  the  amount  of  $535.20  for  2  yr.  4  mo.  18  da. 
at  5  per  cent,  simple  interest. 

10.  Give  formula  or  rule  for  finding  the  base  when  rate 
per  cent  and  difference  are  given.  Form  and  write  such  a 
problem. 


INTEBEST   AND   DISCOUNT.  371 

483.  1.   Find  the  interest  of   $263.75  for  1  yr.,  3  mo. 

16  da.  at  5%. 

2.  Make  a  30-day  bank  note  dated  Jan.  20,  1896,  for 
$600,  payable  at  some  bank.  Find  the  date  of  maturity, 
the  discount,  and  proceeds  if  discounted  on  the  date  of  the 
note.  (Make  the  note  on  a  separate  piece  of  paper,  and 
have  it  properly  indorsed.) 

3.  What  is  the  present  worth  of  a  debt  of  $  500  due  in 
1  yr.  6  mo.,  money  being  worth  6%  ? 

4.  In  what  time  will  $  600  gain  $ 30  interest  at  6%  ? 

5.  What  will   $300  amount  to  in  4  years  compounded 
annually  at  4%  ? 

6.  An  agent  says  he  will  insure  your  house  for  3  years 
at  65.     What  does  he  mean  by  "  at  65  "  ? 

7.  Define  interest ;  principal ;  usury ;  compound  interest. 

8.  Find  the  amount  of  $684.50  for  3  yr.  4  mo.  at  7%. 

9.  Compute  the  interest  of  $  1250  for  2  yr.  5  mo.  12  da. 
by  the  six  per  cent  method. 

10.  What  is  the  interest  on  a  note  for  $515.62,  dated 
March  1,  1885,  and  payable  July  16,  1888? 

484.  1.  A  note  for  $710.50,  with  interest  after  3  mo. 
at  8%,  was  given  Jan.  1,  1884,  and  paid  Aug.  13,  1886. 
What  was'  the  amount  due  ? 

2.  What  sum  of  money  will  gain  $173.97  in  4  yr.  4  mo. 
at  6%  ? 

3.  What  is  the  legal  rate  of  interest  in  this  State  ? 

4.  Find  the  exact  interest  of  $950  at  5%  for  98  days. 

5.  What  principal  will  amount  to  $  1531.50  in  1  yr.  3 
mo.  6  da.  at  6%  ? 

6.  At  what  rate  will  $1500  amount  to  $1684.50  in  2 
years,  18  days  ? 


372  TOPICAL  REVIEW. 

7.  In  what  time  will  $  840  gain  $  78.12  at  6%  ? 

8.  How  long  will  it  take  any  sum  of  money  to  double 
itself  at  4%  ? 

9.  Find  the  compound  interest  of  $460  for  1  yr.  5  mo. 
24  da.  at  6%  interest,  payable  semi-annually. 

10.  If  |-  of  an  acre  of  land  costs  $  15,  what  will  10|-  acres 
cost? 

485.  1.  Name  four  different  forms  of  reduction  of  com- 
mon fractions.  Illustrate  one  of  them  to  show  that  the 
value  of  the  fraction  remains  unchanged. 

2.  Define  Simple  Interest,  True  Discount,  and  Bank 
Discount.  How  does  bank  discount  differ  from  interest? 
How  does  it  differ  from  true  discount  ? 

3.  Define  cancellation,  and  state  the  principle  of  arith- 
metic that  authorizes  its  use. 

4.  Find  the  amount  of  $  575.871  at  5  per  cent,  simple 
interest,  from  Aug.  5,  1883,  to  March  17,  1885. 

5.  What  principal  will  earn  $  71.68  in  2  years,  4  months, 
at  6  per  cent,  simple  interest. 

6.  At  what  rate,  simple  interest,  will  $  175  amount  to 
$  203.35  in  3  yr.  7  mo.  6  days  ? 

7.  In  what  time  will  f  4260  earn  $873.30,  at  6  per 
cent  ? 

8.  A  60-day  note  for  $  610.25,  dated  June  12,  1889,  was 
discounted  in  bank,  July  1,  at  6  per  cent.  Find  the  term  of 
discount,  discount,  and  proceeds. 

9.  Having  purchased  a  horse  for  $  125,  you  wish  to 
borrow  that  amount  at  bank  for  6  mo.  Write  your  own 
note,  indorsed  by  your  parent  as  security,  for  the  sum 
Avhich,  discounted  to-day,  will  give  $  125  as  proceeds  of  the 
note. 


INTEREST   AND  DISCOUNT.  373 

10.  A  stock  of  goods  was  owned  by  three  parties.  A 
owned  |,  B  |,  and  C  the  remainder.  The  goods  were  sold 
at  a  profit  of  $  4260.    What  was  each  one's  share  of  the  gain  ? 

486.  1.  A  horse  is  offered  me  for  $350  cash,  or  for 
$  382.50  to  be  paid  in  4  mo.  What  can  I  save  by  paying 
cash,  the  rate  of  interest  being  6%  ? 

2.  Which  is  the  more  profitable,  and  how  much,  money 
being  worth  5%,  to  buy  a  house  for  $5940  on  2  years' 
credit,  or  for  $  5219.30  on  6  months'  credit  ? 

3.  A  note  dated  June  20,  1893,  and  bearing  interest  at 
6  per  cent,  was  paid  Aug.  15,  1895.  The  face  of  the  note 
being  $  68.45,  what  was  the  amount  paid  ? 

4.  Bought  150  front  feet  of  land  at  f  40  per  front  foot, 
paid  $  116  city  taxes,  $  32  county  taxes,  and  $  320  local 
taxes.  At  the  end  of  two  years  I  sold  for  $  60  per  front 
foot.  Reckoning  interest  at  6%  on  the  purchase  price,  did 
I  gain  or  lose  by  the  transaction,  and  how  much  ? 

5.  A  man  wishes  to  pay  me  $  3252.56.  Not  having  the 
money,  he  borrows  it  from  a  bank  by  giving  his  note  for  48 
days  at  4%.  For  what  sum  does  he  draw  the  note?  No 
grace. 

6. 

$545.50  Bufealo,  N.Y.,  Apr.  2,  1896 

Sixtt/  days  after  date,  I  promise  to  pay^^,,,,,,,^^^^ 

Henry  Hamilton or  order,  Five  hundred  forty - 

jive  and  ^  Dollars.     Value  received. 

Chas.  C.  Trowbridge, 
This  note  was  discounted  May  4,  1896.     Find  the  proceeds. 


374  TOPICAL  REVIEW. 

7.  Required  the  simple  interest  and  amount  of  $  7231.289 
for  3  yr.  8  mo.  15  days  at  8%. 

8.  Face  of  a  note  $750.  Time  60  da.  Eate  6%.  To 
find  proceeds. 

9.  Write  the  following  in  a  note  properly,  and  find  the 
maturity  and  proceeds :  Face,  $  600 ;  date,  April  3,  1896 ; 
due  in  90  days;  discounted  at  bank,  May  20,  1896,  at  6%, 
with  grace. 

lO. 

$9000  Saratoga  Springs,  N.Y.,  Oct.  3,  1895 

Nine  months  after  date,  /  promise  to  pay  to  the 

order  of G-ates  ^  0(9._______iV"me  thousand 

Dollars,  at  the  First  National  Bank.     Value  received. 

S.  B.  Graves. 
Find  the  proceeds,  if  discounted  at  6%,  Dec.  3,  1895. 

STOCKS  AND  AVERAGE  OF  PAYMENTS. 

487.  1.  How  many  shares  of  stock  at  80  can  I  buy  for 
$  2550. 

2.  I  sold  two  houses  for  $  2400  each.  On  one  I  gained 
10%,  on  the  other  I  lost  10%.  How  much  did  both  cost 
me?     Did  I  gain  or  lose  in  the  whole  trade,  and  how  much  ? 

3.  Find  the  cost  of  40  shares  of  American  Express  Co. 
stock  at  105|,  brokerage  i%. 

4.  A  mining  company  declares  a  dividend  of  8%  per 
annum  on  its  stock.  What  is  the  nominal  value  of  a  man's 
shares  who  gets  $  864  as  his  semi-annual  dividend  ? 

5.  If  the  stock  of  a  railway  company  sells  at  5%  above 
par,  what  will  25  shares  cost  ? 

6.  If  I  invest  $21,008  in  5%  bonds  at  104,  what  will 
be  my  annual  income  ? 


STOCKS   AND   AVERAGE  OF   PAYMENTS.  375 

7.  Sugar  bought  at  5  cents  a  pound  was  sold  for  6^ 
cents.     What  per  cent  was  gained  ? 

8.  What  sum  invested  in  4  per  cent  stock  will  yield  an 
annual  income  of  $  320,  if  the  stock  is  purchased  at  par  ? 

9.  What  would  be  the  investment,  if  the  stock  is  worth 
15  per  cent  above  par. 

10.  A  man  invested  his  money  in  6%  railroad  stocks,  and 
received  $  300  semi-annually.    What  was  the  sum  invested  ? 

488.  1.  What  sum  must  be  invested  in  stocks  bearing 
6^  per  cent  interest,  at  105  per  cent,  to  produce  an  annual 
income  of  $  1000  ?     Solve  by  cancellation. 

2.  Define  brokerage,  certificate  of  stock,  par  value,  pre- 
mium (as  used  in  stocks  and  investments).  What  are 
bonds  ?    Name  some  of  the  different  classes  of  bonds. 

3.  What  income  will  be  realized  from  investing 
$4190.63  in  5%  stock,  purchased  at  7%  discount,  if  I  pay 
■J-%  for  brokerage? 

4.  What  is  the  value  of  31  shares  of  $  500  each,  sold  at 
a  premium  of  2^^-^%  ? 

5.  Which  is  more  profitable,  to  buy  8%  bonds  at  25% 
premium,  or  6%  bonds  at  10%  discount  ? 

6.  A  owes  B  $3000,  due  as  follows:  June  15,  $1500; 
Sept.  10,  $400;  Nov.  1,  $500;  Dec.  15,  $600.  B  accepts 
in  settlement  Oct.  26  a  note  for  9  mouths,  bearing  interest 
at  6%  for  the  amount  of  the  debt,  with  6%  interest  due 
him  at  that  date.     Find  the  face  of  the  note. 

7.  On  Jan.  1,  1895,  a  merchant  gave  three  notes :  one 
for  $  500,  payable  in  30  days  ;  one  for  $  400,  payable  in  60 
days ;  and  one  for  $  600,  payable  in  90  days.  What  is  the 
average  term  of  credit,  and  what  the  equated  time  of  pay- 
ment? 


376  TOPICAL   REVIEW. 

8.  E.  R.  Smith  owes  J.  D.  Wilson  $2500,  due  Oct.  12, 
1896.  If  Mr.  Smith  pays  $  500  Aug.  10,  and  $  1000  Sept. 
25f  when  should  the  balance  be  paid  ? 

9.  A  speculator  bought  N.  Y.  C.  stock  at  98^,  and  sold 
it  at  97f,  and  lost  $187.50.  How  many  shares  did  he 
handle  ? 

10.  Had  he  retained  his  stock  until  a  quarterly  divi- 
dend was  declared,  his  dividend  would  have  been  $312.50. 
What  was  the  annual  rate  of  dividend  ? 

489.  1.    State  why  securities  fluctuate  in  value. 

2.  Name  a  corporation. 

3.  What  does  a  stockholder  hold  to  show  that  he  has 
stock  in  a  company  ? 

4.  On  what  does  the  income  from  his  stock  depend  ? 

5.  Why  does  a  corporation  issue  bonds  ? 

6.  Pind  the  present  worth  and  true  discount  of  $  300, 
due  in  10  months,  at  6%. 

7.  Find  the  bank  discount  and  proceeds  of  a  note  of 
$730,  due  in  3  months,  at  6%. 

8.  What  is  the  face  of  a  note  at  2  months  and  18  days, 
which  yields  $  2961  when  discounted  at  a  New  York  bank  ? 

9.  A  person  owning  |  of  a  piece  of  property,  sold  20% 
of  his  share.     What  part  did  he  then  own  ? 

10.  At  what  price  should  4|-%  bonds  be  bought  to  make 
the  income  from  the  investment  equivalent  to  that  from  3% 
bonds  at  par  ? 

PROPORTION  AND  PARTNERSHIP. 

490.  1.   What  is  ratio? 

2.  Eead  the  following :  3 :  15.     What  does  it  equal  ? 

3.  What  is  each  of  the  numbers  in  the  above  expression 
called? 


PEOPORTION  AND  PARTNERSHIP.  377 

4.  What  is  a  proportion  ? 

5.  Is  the  following  expression  a  proportion  ?  Explain 
why.     9  :  12  :  :  16  :  24. 

6.  24  :  (     )  =56:7.     Find  the  omitted  term. 

7.  If  8  men  can  do  a  piece  of  work  in  10  days,  in  how 
many  days  can  12  men  do  it  ? 

8.  If  3  men  in  12  days  of  10  hours  each  can  build  a 
wall  100  feet  long,  14  feet  high,  and  3  feet  thick,  how  long 
will  it  take  4  men  working  8  hours  a  day  to  build  a  wall 
200  feet  long,  16  feet  high,  and  4  feet  thick  ? 

9.  If  it  takes  5  men  4  hr.  24  min.  to  manufacture  400 
boxes,  how  much  time  will  8  men  require  to  perform  the 
same  work. 

10.  If  -f-  of  an  acre  of  land  cost  $  15,  what  will  10|-  acres 
cost  ? 

491.  1.  50  men  in  7  da.  at  12  hours  a  day  dig  a  cellar. 
How  many  men  will  be  required  to  dig  a  similar  cellar  in 
21 J  da.  of  8  hr.  each  ? 

2.  A  and  B  enter  into  partnership,  A  with  $1800  and 
B  with  1^900.  After  8  mo.  B  adds  $300  to  his  capital. 
Divide  a  profit  of  $840  between  them  at  the  end  of  the 
year. 

3.  A  bankrupt  owes  A  $350,  B  $680.50,  C  $65,  D 
$500,  E  $980.50;  his  property  nets  $1648.64.  How  much 
does  each  creditor  receive  ?  How  much  does  he  pay  on  a 
dollar  ? 

4.  What  is  the  ratio  of  7  to  8  ?  Of  2i  to  3i  ?  Of  $  9 
to  $6? 

6.  If  20  men  can  mow  a  field  in  6  days,  in  how  many 
days  will  30  men  mow  it  ? 

6.  If  6  horses  eat  8  bu.  14  qt.  of  oats  in  9  days,  at  the 
same  rate  how  long  will  66  bu.  30  qt.  last  17  horses  ? 


378  TOPICAL  REVIEW. 

7.  A  and  B  hired  a  pasture  for  $  40  for  the  season.  A 
put  in  9  cows  for  4  mo.,  and  B  put  in  8  cows  for  8  mo. 
Other  conditions  being  the  same,  what  should  each  pay  ? 

8.  In  what  time  will  $  10,000  yield  $  1200  interest  at 
8%.     Solve  by  proportion. 

9.  If  the  antecedent  is  |  of  ^  of  ^%,  and  the  ratio  is  f 
of  |4-  of  i|,  what  is  the  consequent  ? 

10.    Required  the  ratio  of  6 J  cu.  ft.  to  11|  cu.  ft. 

492.  1.  A,  B,  and  C  entered  into  partnership.  A  put 
in  $  600  for  8  mo.,  B  |  800  for  7  mo.,  C  $  1500  for  4  mo. 
They  gained  $  820.     What  was  each  one's  share  of  the  gain  ? 

2.  A,  B,  and  C  found  a  gold-mine,  and  after  developing 
it  sold  it  for  $64000.  They  agreed  to  divide  the  money 
according  to  the  time  each  had  worked.  A  had  worked  37 
days,  B  46  days,  and  C  39  days;  for  extra  services  B  is  to 
receive  $  1800,  and  C  $  1200  additional.  How  much  does 
each  receive  ? 

3.  Three  men,  A,  B,  and  C,  enter  into  partnership.  Out 
of  a  gain  of  $  1200,  C  takes  $  500  and  B  $  400.  A's  in- 
vestment is  $  4500.     Find  B's  and  C's  investment. 

4.  Divide  $  450  among  three  people  in  the  ratio  of  3,  4, 
and  8. 

5.  Three  persons  bought  a  block  for  $  21000,  of  which 
A  paid  $  9000,  B  $  8000,  and  C  the  remainder.  They 
rented  it  for  $  1400  a  year.  What  was  each  man's  share  of 
the  rent  ? 

6.  Forster,  Stull,  and  Furlong  made  8000  pairs  of  bi- 
cycle pedals  in  1895,  which  they  sold  for  $  1.60  per  pair. 
The  pedals  cost  them  $  1.15  per  pair.  If  Mr.  Forster  put 
in  $  1000  Jan.  1,  Mr.  Stull  $  1200  April  1,  and  Mr.  Furlong 
$  900  May  1,  what  would  be  each  one's  share  of  the  gain 
after  drawing  out  the  original  investment  ? 


INVOLUTION   AND  EVOLUTION.  379 

7.  Four  men  purchased  a  city  block  for  $  36,000.  The 
first  contributed  $20,000,  the  second  $7000,  the  third 
$  4000,  and  the  fourth  $  5000.  They  sold  the  land  at  an 
advance  of  50%  on  the  purchase  price.  How  much  was 
each  man's  share  of  the  gain  ? 

8.  A,  B,  and  C  form  a  partnership  in  which  A  is  to 
furnish  no  capital,  but  give  his  whole  time  to  the  business, 
and  have  J  the  profits.  B  furnishes  $  10,000,  and  C 
$  15,000.  Their  net  profit  at  the  end  of  a  year  is  $  8000. 
What  is  each  partner's  share  ? 

9.  A,  B,  and  C  gain  in  business  together  respectively 
$  700,  $  1000,  and  $  1500.  What  was  the  investment  of 
each  if  their  joint  capital  was  $  16,000  ? 

10.  Smith,  Brown,  and  Jones  gain  in  trade  $9400. 
Smith  furnished  $  10,000  for  5  months.  Brown  $  9000  for 
6  months,  Jones  $  7000  for  1  year.     Apportion  the  gain. 

INVOLUTION   AND  EVOLUTION. 

493.  1.  Define  involution;  evolution;  a  square;  cube 
root. 

2.  Find  the  square  of  6f ;  of  2.35. 

3.  Find  the  third  power  of  123. 

4.  Find  the  square  root  of  the  fraction  fffj. 

5.  What  is  the  distance  around  a  square  field  which 
contains  40  acres  ? 

6.  A  man  has  640  acres  of  land.  How  much  more  will 
it  cost  to  enclose  it  with  a  fence  at  $  4  a  rod,  in  a  rectangu- 
lar form  512  rods  long  and  200  rods  wide,  than  it  would  if 
in  the  form  of  a  square  ? 

7.  What  is  the  length  of  one  side  of  a  cube  which  con- 
tains 8120601  cubic  inches  ? 

8.  Find  the  entire  surface  of  a  cube  whose  volume  is  42 
cu.  ft.  1512  cu.  in. 


380  TOPICAL   REVIEW. 

9.  The  edge  of  a  cube  is  42  inches.  Find  the  length  of 
the  edge  of  another  cube  4  times  as  large. 

10.  If  16  cords  of  wood  be  piled  in  the  form  of  a  cube, 
what  will  be  the  length  of  one  of  its  edges  ? 

494.  1.  What  are  the  length  and  breadth  of  a  rectangu- 
lar field  which  contains  60  acres,  the  length  of  which  is 
three  times  its  breadth  ? 

2.  A  rectangular  farm  of  300  A.  is  7J  times  as  long  as  it 
is  wide.     How  many  miles  of  fence  will  enclose  it  ? 

3.  A  bird  is  15  feet  above  a  monument  80  ft.  high.  A 
boy  is  145  ft.  from  the  bird.  How  far  is  the  boy  from  the 
base  of  the  monument  ? 

4.  How  far  is  it  between  the  extreme  corners  of  a  box 
10  ft.  square  and  6  ft.  deep  ? 

5.  Eind  how  many  acres  in  a  lot  in  the  form  of  a  right- 
angled  triangle  whose  hypothenuse  is  50  rd.  and  whose  base 
is  40  rd. 

6.  Find  the  diagonal  of  a  square  piece  of  land  equal  in 
area  to  a  rectangular  piece  whose  dimensions  are  80  rd.  by 
20  rd. 

7.  Wishing  to  know  the  height  of  a  church  steeple,  I 
find  it  casts  a  shadow  165  ft.  I  also  find  that  a  10-ft.  pole, 
placed  perpendicularly,  casts  a  shadow  12^  ft.  What  is  the 
height  of  the  steeple  ? 

8.  A  house  is  36  ft.  wide,  and  the  ridge  of  the  roof  is  ^ 
12  ft.  above  the  plates.     How  long  are  the  rafters  ? 

9.  A  steamer  goes  due  north  at  the  rate  of  12  miles 
an  hour,  and  another  goes  due  east  at  the  rate  of  15  miles  an 
hour.     How  far  apart  will  they  be  at  the  end  of  8  hours  ? 

10.  If  a  pineapple  5  in.  in  diameter  costs  20^,  what  should 
be  the  cost  of  a  pineapple  of  similar  shape  6  in.  in  diameter  ? 


MISCELLANEOUS. 


495.  1.  The  sum  of  two  numbers  is  2120,  and  their 
difference  938.     What  is  each  number? 

2.  J.  &  E.  Eoss,  New  York,  bought  of  A.  L.  Covert  & 
Co.,  Philadelphia,  the  following  articles,  June  20,  1881 :  15 
Nichols's  Geography  at  f  0.65;  12  Meiklejohn's  Literature 
at  1 0.80 ;  25  Bowser's  Geometry  at  $  0.75 ;  15  Hawthorne 
&  Lemmon's  Literature  at  $  1.12 ;  10  Thomas's  United  States 
History  at  $  1.00. 

They  paid  $  25  in  cash,  and  returned  books  to  the  amount 
of  $  10.     Make  out  a  bill  showing  the  entire  statement. 

3.  A  city  contains  22,000  inhabitants.  If  each  inhab- 
itant should  contribute  one  cent  per  week  for  fifty-two 
weeks  towards  the  erection  of  a  soldiers'  monument,  how 
expensive  a  monument  could  be  built  at  the  end  of  the 
year  ? 

4.  The  State  of  New  York  has  7746  miles  of  railroad, 
which  cost  $  588,672,762.     Find  the  average  cost  per  mile. 

5.  The  sum  of  three  numbers  is  96:  the  least  is  4|-,  and 
the  greatest  37|.  Find  the  other  number  and  the  product 
of  the  three  numbers. 

6.  $9,000,000  has  recently  been  appropriated  for  improv- 
ing the  Erie  Canal.  If  it  is  352  miles  long,  how  many  dol- 
lars may  be  expended  on  each  mile  ? 

381 


382  MISCELLANEOUS. 

» 

7.  Find  the  least  common  multiple  of  24,  60,  75,  120. 

8.  What  is  the  smallest  sum  of  money  with  which  I 
can  purchase  oxen  at  $  30  each,  cows  at  $  60  each,  or  horses 
at  $  80  each  ? 

9.  Find  the  difference  between  the  greatest  common 
divisor  and  the  least  common  multiple  of  81,  45,  108,  and 
135. 

10.    What  is  the  greatest  number  that  will  exactly  divide 
3640,  12750,  and  18755  ? 

^  496.  1.  If  the  ties  on  the  N.  Y.  C.  &  H.  R.R.  are  If  ft. 
apart  from  centre  to  centre,  how  many  are  there  from  New 
York  to  Buffalo,  a  distance  of  450  miles  ? 

2.  If  the  Empire  State  express  has  an  average  rate  of 
62  miles  an  hour,  how  many  hours  and  minutes  will  it 
take  to  run  from  Syracuse  to  Albany,  a  distance  of  150 
miles  ? 

3.  Multiply  7f  by  17|f . 

4.  E.  C.  Stearns  &  Co.  sell  24  bicycles  at  $  62|-  apiece. 
What  do  they  bring  ? 

5.  How  many  times  does  a  bicycle  wheel  9J  ft.  in  cir- 
cumference revolve  in  going  3  miles,  there  being  5280  ft. 
in  a  mile  ? 

6.  Multiply  i^  +  i  by  121 

^  2    "T"  "3" 

^7.  A  and  B  can  build  a  house  in  30  days :  B  can  do  the 
work  alone  in  45  days.  In  how  many  days  can  A  do  it 
alone  ? 

8.  Write  a  complex  fraction,  whose  numerator  shall  be 
a  simple  fraction,  and  its  denominator  compound. 

9.  A  drover  bought  375  sheep  at  $4|-  per  head.  He 
sold  200  of  them  at  a  loss  of  20  cents  per  head,  and  gained 
enough  on  the  rest  to  balance  the  loss.  What  did  he  receive 
per  head  for  the  rest  ? 


PROBLEMS.  383 

/ 

10.  A  can  do  a  piece  of  work  in  5  days  ;  B  can  do  the 
same  work  in  8  days.  In  what  time  can  they  do  it  working 
together  ? 

497.  1.  A  boy  paid  for  a  book  $.70,  which  was  f  of  his 
money.  The  remainder  he  spent  for  marbles  at  2^  cents 
apiece.  How  much  money  had  he  at  first,  and  how  many 
marbles  did  he  buy  ? 

2.  At  a  school  examination  ^  of  the  pupils  passed,  and 
250  pupils  failed.  How  many  pupils  were  examined,  and 
how  many  passed  ?  n 

3.  ^  of  a  number  diminished  by  -|  of  it  is  equal  to  5. 
What  is  the  number  ? 

4.  ^4_.  of  1743  is  j\\  of  what  number  ? 

5.  A  man  after  giving  ^,  \,  and  -|-  of  his  money  in 
charity  had  $  10000  left.     How  much  had  he  at  first  ? 

6.  Four  persons  own  a  ship.  A  owns  J  of  it,  B  -J-  of 
the  remainder,  C  J  of  what  then  remained,  and  T)  the 
remainder,  which  is  worth  $3000.  What  is  the  value  of 
the  ship  ? 

7.  If  I  of  a  number  be  divided  by  4,  and  -|-  of  \  of  the 
number  be  taken  from  the  quotient,  the  remainder  will  be 
6.     What  is  J  of  the  number  ? 

8.  One  person  can  do  a  piece  of  work  in  6  days,  another 
can  work  twice  as  fast.  How  long  will  it  take  them  to  do 
the  work  together  ? 

9.  A  boy  was  asked  how  many  fish  he  had  caught. 
He  said  that  the  difference  between  ^  and  -|  the  number 
was  six.     How  many  had  he  ? 

10.  A,  B,  and  C  can  do  a  piece  of  work  in  5  da.  A  can 
do  it  alone  in  12  da.,  C  can  do  it  in  15  da.  In  what  time 
can  B  do  it  ? 


384  MISCELLANEOUS. 

498.  1.  What  will  it  cost  at  $1.75  a  yard  to  carpet  a 
floor  18  ft.  long,  14  ft.  wide,  with  carpet  |  yd.  wide  ? 

2.  How  many  yards  of  carpeting  27  inches  wide  will  be 
required  for  a  room  30  ft.  long,  24  ft.  wide,  if  the  strips  run 
crosswise,  and  6  inches  be  allowed  for  matching  ? 

3.  What  fraction  of  a  great  gross  is  3  gross,  5  doz., 
If  units  ? 

4.  At  $  .27  per  square  yard,  find  the  cost  of  plastering  a 
room  30  ft.  by  24  ft.  by  12  ft.  high,  allowing  for  a  base- 
board 1  foot  high,  two  doors  9  ft.  by  3  ft.,  and  5  windows 
6  ft.  by  3  ft. 

5.  Reduce  5  cd.  ft.  9|  cu.  ft.  to  the  fraction  of  a  cord. 

6.  Reduce  33  gal.  3  qt.  1  pt.  1^-^  gi.  to  the  fraction  of 
a  hhd. 

7.  How  much  tin  will  be  required  to  make  a  pail  and 
cover,  the  pail  to  be  6  inches  in  depth  and  7  inches  in 
diameter,  and  the  rim  of  the  cover  to  be  1  inch  deep? 

8.  At  $  16.50  per  M.,  what  will  be  the  cost  of  12  sticks 
of  timber,  each  14  ft.  long,  10  in.  wide,  and  8  in.  thick  ?        A 

9.  How  many  board  feet  in  a  plank  16  ft.  long,  15  in. 
wide  at  one  end  and  10  in.  wide  at  the  other  end,  and  3  in. 
thick  ? 

10.  The  longitude  of  New  York  is  74°  0'3"  W.,  and  that 
of  San  Francisco  122°  23'  W.  When  it  is  1  p.m.  at  New 
York,  what  is  the  time  at  San  Francisco  ? 

499.  1.  The  longitude  of  Syracuse,  N.Y.,  is  76°  9'  16" 
W.,  and  that  of  Berlin,  Germany,  is  13°  23'  44"  E.  When 
it  is  noon  in  Berlin,  what  is  the  time  at  Syracuse  ? 

2.  The  Oswego  River  is  24  miles  long,  and  descends 
120  feet  in  that  distance.  What  is  the  average  descent 
per  mile  ? 


XTNITERSIXr 

•S££AUF025^ 
PROBLE>fS.  385 

3.  Add  I  A.,  ^  sq.  rd.,  J  sq.  yd.,  |  sq.  ft. 

4.  Find  the  cost  of  4  T.  7  cwt.  40  lb.  of  hay  at  $  12  per 
ton. 

5.  From  a  cask  containing  44  gal.  2  qt.  1  pt.  of  vine- 
gar, 8  gal.  3  qt.  leaked  out.  What  decimal  of  the  original 
contents  remained? 

6.  Find  the  number  of  square  inches  in  the  surface  of 
a  block  2  ft.  long,  18  in.  wide,  and  10  in.  high. 

7.  The  sun  rose  in  the  latitude  of  New  YoVk,  April  1, 
1896,  at  5  o'clock  and  43  minutes,  and  set  at  6  o'clock  and 
25  minutes.  It  rose  April  30  at  4  o'clock  and  59  minutes, 
and  set  at  6  o'clock  and  55  minutes.  How  much  longer 
was  the  thirtieth  day  than  the  first  ? 

8.  How  long  and  wide  must  a  granary  be  to  hold  4000 
bushels  of  grain,  if  it  is  8  ft.  high,  and  the  grain  to  be 
placed  in  bins  6  ft.  back  on  each  side  of  an  aisle  4  feet 
wide  ? 

•     9.    A  cubic  ft.  of  water  weighs  62-1-  pounds.     How  many 
barrels  in  a  cistern  of  water  that  weighs  6  T.  5  cwt.  ? 

10.    Find  the  cost  of  1  bu.  1  pk.  1  qt.  and  1  pt.  of  chest- 
nuts at  5  ^  per  quart. 

500.    1.    At  what  rate  per  cent  will  $  2500  gain  $  625  in 
3  years,  4  months  ? 

2.  A  merchant  buys  goods  at  $  1.20  a  yard,  and,  after 
keeping  them  6  mo.,  sells  them  at  $  1.35.  What  is  his  rate 
of  gain  ? 

3.  A  man  buys  oranges  at  1  ^  each,  and  sells  them  at 
18  cents  a  dozen.     What  is  his  gain  per  cent  ? 

4.  Find  the  amount  on  $  836.22  from  Feb.  19,  1895,  to 
June  3,  1896,  at  6%. 

5.  3200  votes  are  cast  for  two  men;  one  has  a  majority 
of  374.     How  many  votes  did  each  receive  ? 


386  MISCELLANEOUS. 

6.  A  man  borrowed  $756.12,  June  28,  1872.  What 
must  he  pay  to  cancel  the  debt  July  11,  1872,  at  6%  ? 

7.  A  commission  merchant  in  Minneapolis  received 
$  6150,  with  directions  to  purchase  flour.  His  terms  were 
2^%  on  the  amount  purchased.  How  many  barrels  of 
flour  at  $  3  a  barrel  can  he  ship  to  the  sender  of  the 
money  ? 

8.  A  merchant  sells  goods  at  an  advance  of  20%,  but 
loses  5%  of  his  sales  by  bad  debts.     What  %  does  he  gain  ? 

9.  A  bought  a  carriage  at  20%  discount  with  10%  and 
5%  off,  and  sold  it  at  the  list  price.  What  %  profit  did  he 
make  ? 

10.  An  agent  sold  some  Western  land,  and  paid  to  the 
former  owner  $  7531.30,  retaining  f  153.70  as  commission. 
What  rate  did  he  charge  ? 

501.  1.  A  district  schoolhouse  cost  $8010;  the  valua- 
tion of  the  property  of  the  district  is  $392,375,  and  the 
number  of  polls  assessed  at  $  1.25  each  is  130.  What  is 
the  rate  of  tax,  and  what  was  A's  tax,  who  paid  for  4  polls, 
the  valuation  of  his  property  being  $  6000  ? 

2.  What  sum  of  money  placed  on  interest  at  6%  will 
amount  to  $  1567.85  in  1  year,  3  months  ? 

3.  Sold  wheat  at  72  cents  per  bushel,  and  thereby  lost 
10%  of  the  cost.     What  was  the  cost  per  bushel  ? 

4.  What  will  be  the  net  cost  of  stationery  billed  at 
$  850,  if  the  discount  is  20%  and  10%  off  ? 

5.  A  house  worth  $  7200  is  insured  for  -|  of  its  value,  at 
the  rate  of  60  cents  on  $  100.     Find  the  premium. 

6.  A  man  sold  a  house  for  $  4200,  which. was  20%  more 
than  it  cost  him.     What  did  it  cost  ? 


PROBLEMS.  387 

« 

7.  On  a  bill  of  goods  listed  at  $645,  choice  is  given 
between  discounts  of  20%,  10%,  and  5%  off,  or  a  direct     V 
discount  of  35  %  off.     Which  is  better,  and  how  much  ? 

8.  If  a  merchant  gains  16|%  by  selling  cloth  at  $1.40 
per  yard,  find  his  gain  on  a  sale  amounting  to  $  32. 

9.  I  owe  B  a  bill  of  $  1980.  If  I  borrow  the  money 
from  a  bank,  what  must  be  the  face  of  a  note,  due  in  60 
days  without  interest,  which  I  must  give  to  the  bank,  that 
I  may  receive  the  amount  necessary  to  pay  him,  discount 
at  6%  ? 

10.  A  man  sells  his  house  for  $8000,  and  receives  in 
payment  a  note  for  90  days.  After  30  days  he  has  the  note 
discounted  at  a  bank  at  6%.     What  does  he  receive  for  it  ? 

502.  1.  I  was  offered  $  160  cash  for  my  buggy,  or  a  note 
of  $  165  payable  in  90  days.  I  took  the  note,  and  dis- 
counted it  at  a  bank  at  5%.  Did  I  gain  or  lose,  and  how 
much  ? 

2.  What  is  the  difference  between  the  true  and  bank 
discount  on  $  1250  for  90  days  at  6%  ? 

3.  If  John  lends  James  $  300  for  4  months,  how  long 
ought  James  to  lend  John  $  800  to  equal  the  favor  ? 

4.  I  have  a  note  of  $  1225,  due  in  48  days.  Needing 
the  money  immediately,  I  get  it  discounted  at  a  bank  at 
6  % .  How  kauch  shall  I  receive,  and  how  much  will  the 
bank  take  ?%  No  grace. 

5.  Three  men  hire  a  pasture  for  $  60.  A  put  in  4  cows  "w 
for  11  weeks,  B  5  cows  for  12  weeks,  and  C  8  cows  for  5  A 
weeks.     What  ought  each  to  pay  ? 

6.  If  a  man  5  ft.  10  in.  high  casts  a  shadow  4  ft.  6  in. 
long,  what  is  the  height  of  a  tree  which  casts  a  shadow  85 
ft.  long  at  the  same  time  ? 


X 


388  MISCELLANEOUS. 

7.  Give  the  inverse  ratio  of  -^  to  ^- 

8.  Required  the  ratio  of  £  21  15s.  to  £  6  IBs. 

9.  A,  B,  and  C  entered  business  with  a  certain  capital, 
Jan.  1,  1894.  Jan.  1,  1896,  they  find  the  business  to  be 
worth  $  7000,  which  is  a  gain  of  40%  on  the  original  capital. 
A's  share  of  the  gain  is  50%,  B's  share  30%,  and  C's  share 
20%.     What  amount  did  each  invest  ? 

10.    What  did  each  gain  in  Example  9  ? 

503.  1.  If  4  barrels  of  flour  will  last  three  persons  for  1 
year,  how  many  barrels  will  be  required  to  last  10  persons 
10  months  ? 

2.  The  shadow  of  a  flag-staff  at  a  certain  time  of  day 
was  64  feet  in  length.  A  line  stretched  from  the  top  of  the 
flag-staff  to  the  extremity  of  the  shadow  measured  150  feet. 
Required  the  height  of  the  staff. 

3.  Messrs.  Stevens,  Jones,  &  Payne  form  a  partnership, 
placing  into  their  business  $  350,  $  450,  $  1500  respectively. 
They  make  $  570  the  first  year.  What  share  of  the  profits 
should  each  receive  ? 

4.  By  selling  3%  stock  at  par,  and  buying  4%  stock  at 
110,  a  man  increases  his  income  $  105  a  year.  How  many 
shares  of  the  3  %  stock  does  he  sell  ? 

5.  A,  B,  and  0  enter  into  a  partnership.  A  furnished 
$  1200  for  8  mo.,  B  furnished  $  1600  for  9  mo.,  and  C  fur- 
nished $  1000  for  a  year.  They  lose  $  560.  What  is  each 
man's  loss  ? 

6.  What  is  the  length  of  a  walk  laid  diagonally  through 
a  park  which  measures  60  rods  on  one  street  and  80  rods  on 
another  ? 

7.  What  will  be  the  difference  in  ratio  of  income  between 
6%  stock  bought  at  120  and  4%  bought  at  95  ? 


PROBLEMS.  389 

8.  A  fatKer  dying  left  to  his  family  a  certain  sum  of 
money,  of  which  the  wife  received  $  8000,  his  daughter 
$  4000,  and  each  of  two  sons  $  6000.  What  part  of  the 
whole  did. each  receive  ? 

9.  If  sugar  costs  5^  cents  per  pound  and  coffee  33  cents 
per  pound,  what  is  the  ratio  of  the  cost  of  the  sugar  to  that 
of  the  coffee  ? 

10.  At  $.50  per  rod,  how  much  will  it  cost  to  enclose  a 
field  of  80  acres,  that  is  twice  as  long  as  it  is  wide  ? 

504.  1.  If  the  sale  of  coal  at  $  .75  per  ton  above  cost 
yields  a  profit  of  18|%,  how  much  must  the  seller  add  to 
this  price  to  make  a  profit  of  40%  ? 

2.  At  what  price  must  a  4%  stock  be  purchased  to  yield 
5%  on  the  investment? 

3.  If  a  pile  of  wood  32  ft.  long,  4  ft.  wide,  and  4  ft.  high, 
costs  $  32.50,  what  will  be  the  cost  of  a  pile  64*ft.  long,  8  ft. 
wide,  and  8  ft.  high  ? 

4.  If  10  men,  working  10  hr.  a  day  for  30  da.,  can  build 
a  fence  200  rd.  long,  how  many  men,  working  6  hr.  a  day  for 
10  da.,  can  build  92  rd.  of  the  same  kind  of  fence  ? 

5.  The  smaller  of  two  numbers  is  36,  and  one-half  of  the 
ratio  between  it  and  the  larger  is  2.  What  is  the  larger 
number? 

6.  What  number  has  the  same  ratio  to  5  that  ^  has  to  J  ? 

7.  Find  the  mean  proportional  between  16  and  36. 
Between  -^^  and  1. 

8.  What  income  on  his  investment  will  a  man  realize  if 
he  purchases  4%  stock  at  125  ? 

9.  If  A's  capital  is  $  3000,  and  B's  $  2000,  how  much 
more  should  B  invest  at  the  end  of  6  months  that  he  may 
share  equally  with  A  at  the  end  of  the  year  ? 

10.  What  is  the  rate  per  cent  of  a  tax  for  $52.88J  on 
property  assessed  at  $  3525.50  ? 


390  MISCELLANEOUS. 

505.  1.  Write  your  own  promissory  note  for  $  200,  with 
interest  payable  in  90  days  from  to-day  to  any  person  you 
choose. 

2.  On  what  month  and  day  would  your  note  become  due, 
including  days  of  grace  (Ex.  1)?  Give  one  reason  why  the 
note  is  void  (worthless).  Find  the  amount  due  on  your  note 
at  its  maturity. 

3.  What  is  the  time  of  day  when  the  time  past  noon 
equals  the  time  to  midnight?  When  i  the  time  past 
noon  equals  the  time  to  midnight?  When  the  time  past 
noon  equals  J  the  time  to  midnight  ? 

4.  A  cask  can  be  emptied  by  a  i-inch  faucet  in  4  hours. 
In  what  time  can  it  be  emptied  by  a  li-inch  faucet  ? 

5.  Explain  the  difference  between  factor  and  root;  be- 
tween product  and  power. 

6.  A  and  B  divide  $90  in  the  rafio  of  |  to  |.  What  is 
each  one's  share  ? 

7.  If  a  tank  131  ft.  long,  7 J  ft.  wide,  and  3i  ft.  deep 
holds  73J  barrels  of  water,  how  wide  must  another  tank  be 
that  is  9  ft.  9  in.  long,  4  ft.  10  in.  deep,  and  holds  89^ 
barrels  ? 

8.  l  +  (|y-</A^. 

7  X  (if 

9.  A  milkman's  quart  measure  is  too  small  by  one  gill. 
At  5  cents  a  quart,  how  much  does  he  dishonestly  make  in 
the  month  of  June,  if  he  sells  500  false  quarts  daily  ? 

10.  Find  the  length  of  the  diagonal  of  an  are  of  land  in 
the  form  of  a  square. 


MENSURATION. 


506.  The  process  of  measuring  lines,  surfaces,  and  solids 
is  Mensuration. 

507.  A  Line  is  that  which  has  length,  without  breadth 
and  thickness. 

508.  A  Straight  Line  is  the  shortest  distance  between  two 
points,  or  a  line  that  does  not  change  its  direction  at  any 
point. 

509.  A  Curved  Line  changes  its  direction  at  every  point. 

510.  A  Plane  Surface  is  a  surface  that  does  not  change 
its  direction.  f 

511.  A  Quadrilateral  is  a  plane  figure  having  four  straight 
sides. 

512.  Parallel  Lines  are  lines  having  the  same  direction 
and  equally  distant  from  each  other. 

513.  A  Parallelogram  is  a  quadrilateral  whose  opposite 
sides  are  parallel. 

Ir     514.   A   Rhomboid  'is  a  parallelogram  whose   angles  are 
not  right  angles. 

What  is  a  parallelogram  called  whose  angles  are  right 
angles  ? 

391 


392  MENSURATION. 

''    515.   A  Rhombus  is  a  rhomboid  whose  sides  are  equal. 


A   Rhomboid.  A   Rhombus. 

y/    516.    The  area  of  a  rhomboid  is  found  by  multiplying  the 
base  by  the  altitude. 

Note.  —  The    altitude    of   a   parallelogram    is    the   perpendicular 
distance  between  the  sides. 

1.  Find  the  area  of  a  parallelogram  whose   base   is  24 
rods,  and  altitude  18  rods. 

2.  Find  the  area  of  a  rhombus  whose  base  is  15  ft.  and 
altitude  8  ft. 

3.  Draw  a  rhomboid  whose  base   is  15  ft.  and  altitude 
10  ft.     Find  its  area. 

Draw  a  rectangle  having  the  same  dimensions. 

1/  517.    A  Trapezoid  is  a  quadrilateral  having  only  two  sides 
parallel. 


518.  To  find  the  area  of  a  trap- 
ezoid, multiply  \  the  sum  of  the 
parallel  sides  by  the  altitude. 

4.  Find  the   area  of  a  trap-  a  Trapezoid. 
ezoid  whose  altitude  is  10  ft.,  its 

longest  side  20  ft.,  and  shortest  side  15  ft. 

5.  A  board  20  inches  wide  at  one  end  and  12  inches 
wide  at  the  other  is  16  feet  long.     How  many  board  feet 

does  it  contain?  y^^^^»^_ ->  ^    4,  3£i^^  "^     J--    U\^,  "^  ^ 

.^  -       . 

519.  A  Trapezium  is  a  quadrilateral  having  no  two  sides 
parallel. 


SOLIDS. 


393 


Note.  —  By  drawing  a  diagonal  between  any  two  opposite  sides  of 
a  trapezium,  we  have  two  triangles,  the  diagonal  serving  as  the  base  of 
each.  The  altitude  of  each  is  the 
perpendicular  distance  from  its  other 
angle  to  the  diagonal. 

V  520.  To  find  the  area  of  a 
trapezium,  multiply  the  diag- 
onal by  half  the  sura  of  the 
altitudes  of  the  two  triangles.  ^  Trapezium. 

6.  The  diagonal  of  a  trapezium  is  18  ft.;  the  altitudes 
of  its  two  triangles  are  5  ft.  and  3  ft.     What  is  the  area  ? 

7.  A  farm  is  in  the  form  of  a  trapezium.  The  diagonal 
distance  between  the  northern  and  southern  corners  is  108 
rods,  and  the  perpendicular  distances  from  the  east  and 
west  corners  to  the  diagonal  are  52  rods  and  36  rods 
respectively.     How  many  acres  in  the  farm? 


SOLIDS. 

621.  A  solid  whose  two  bases  are  equal  and  parallel,  and 
its  other  faces  parallelograms,  is  called  a  Prism. 

Note. — Prisms  take  their  names  from  the  form  of  their  bases,  as 
triangular,  quadrangular,  pentagonal,  hexagonal,  etc.,  according  as 
the  bases  have  three,  four,  five,  or  six  sides,  etc. 

^  522.  To  find  the  contents  of  a 
prism,  multiply  the  area  of  the 
base  by  the  altitude. 

8.  Find  the  contents  of  a  tri- 
angular prism  whose  altitude  is 
10  in.,  and  area  of  base  7  sq.  in. 

9.  What  are  the  contents  of 
a    quadrangular    prism   whose 

base  is  5  in.  by  8  in.,  and  whose  altitude  is  12  in.  ? 

10.  What  are  the  contents  of  a  hexagonal  prism,  the  area 
of  whose  base  is  10  sq.  ft.,  and  whose  altitude  is  15  ft.  ? 


A  Triangular 
Priam. 


A  Rectangular 
Prism. 


394  MENSURATION. 

PYRAMIDS    AND    CONES. 

y  523.  A  solid  whose  base  is  a  triangle,  square,  pentagon, 
etc.,  and  whose  sides  are  triangles  meeting  at  a  vertex,  is 
called  a  Pyramid. 

Note.  — A  pyramid  takes  its  name  from  the  form  of  its  base. 

t/     A  solid  whose  base  is  a  circle,  and  whose  convex  surface 
terminates  in  a  point,  is  called  a  Cone. 

^                         .  524.    The  Altitude  of  a  pyr- 

Af\^  amid  or  cone  is  the  perpendic- 

c*/  I  m  ^"^^^  distance  from  its  vertex 

^ii  '  m  *^  *^®  centre  of  its  base. 

#^^™  The    Slant    Height    is    the 

D      ^l.j4m^v^  shortest    distance    from     the 

^     X     C  vertex    to    the    perimeter   of 


A  Pyramid.  A  Cone. 


the  base. 


r..h^ 


525.  To  find  the  contents  of  a  pyramid  or  cone,  multiply 
the  area  of  the  base  by  \  of  the  altitude. 

To  find  the  convex  surface,  multiply  the  perimeter  of  the 
base  by  \  the  slant  height. 

11.  Find  the  contents  of  a  quadrangular  pyramid  whose 
altitude  is  40  in.,  and  whose  sides  of  bases  are  8  in.  and 

6  in. 

Solution,  — 8  x  6  x  ^3^  =  160  cu.  in.     Ans. 

12.  Find  the  convex  surface  of  a  regular  hexagonal 
pyramid  whose  slant  height  is  16  in.,  and  whose  side  of 
base  is  4  in. 

13.  Find  the  convex  surface  of  a  cone,  when  the  circum- 
ference of  its  base  equals  16  ft.  and  its  slant  height  18  ft. 

14.  Find  tlie*Convex  surface  and  volume  of  a  cone  whose 
radius  is  4  in.  and  altitude  6  in. 

526.  The  Frustum  of  a  cone  or  pyramid  is  the  part  which 
is  left  after  the  top  is  cut  off  in  a  plane  parallel  to  the  base. 


SOLIDS 


395 


Frustum  of  a  Pyramid.         Frustum  of  a  Cone. 


527.  To  find  the  contents  of  the  frustum  of  a  pyramid  or 
cone,  multiply  ^  of  the  altitude  by  the  sum  of  the  areas  of 
the  two  bases  plus  the 
square  root  of  their 
product. 

15.  Find  the  con- 
tents of  the  frustum  of 
a  quadrangular  pyra- 
mid whose  altitude  is 
15  ft.,  and  whose  ends 
are  6  ft.  and  4  ft.  square. 

16.  A  log  16  ft.  long  is  30  in.  in  diameter  at  one  end  and 
24  in.  at  the  other.     Find  its  cubical  contents. 

528.  A  Sphere  is  a  solid  bounded  by  a  curved  surface,  all 
parts  of  which  are  equally  distant  from  the  centre. 

529.  To  find  the  surface  of  a  sphere, 

multiply    the    circumference    by    the 
diameter.  ^  ^ 

530.  To  find  the  contents  of  a  sphere, 

multiply  the  surface  by  ^  of  the  diameter. 

17.  Find   the   surface   of    a   sphere 
when  the  diameter  is  16  inches. 

18.  Find    the    surface    of    a   sphere 
when  the  radius  is  3  yd. 

19.  Find  the  surface  of  a  sphere  when  the  radius  is  5  cm. 

Find  the  volume  when : 

20.  Diameter  =  25  ft.      22.    Radius  =  12  ft. 

21.  Radius  =  2  ft.  23.    Circumference  =  12.5664  in. 

24.  Radius  =  3  dm. 

25.  Compare  the  volume  of  a  4-ft.  cube  and  a  4-ft.  sphere. 

26.  Compare  the  surfaces  of  a  4-ft.  cube  and  a  4-ft.  sphere- 


A  Sphere. 


X 


396  MENSURATION. 

REVIEW    OF   MENSURATION. 

531.  1.  Find  the  cubic  yards  in  a  cone,  the  circumference 
of  whose  base  is  20  ft.,  and  whose  altitude  is  30  ft. 

2.  Find  the  area  of  a  semi-circle  when  its  radius  equals 
14  ft. 

3.  Find  the  area  of  a  square  inscribed  in  a  circle  of  4  ft. 
in  diameter. 

4.  The  circumference  of  a  circle  and  the  perimeter  of  a 
square  are  each  300  ft.     Which  has  the  greater  area  ? 

5.  A  circle  is  inscribed  in  a  6-ft.  square.  Find  the  area 
of  the  circle. 

6.  Find  the  value  at  $50  an  acre  of  a  farm  in  the  form 
^  of  a  trapezoid,  the  parallel  sides  of  which  are  120  rd.  and 

160  rd.  respectively,  the  distance  between  which  is  80  rd. 

7.  How  many  miles  does  the  earth  travel  in  a  revolu- 
tion around  the  sun,  the  distance  between  them  being 
95,000,000  miles  ? 

8.  If  a  bin  is  8  feet  square,  how  deep  must  it  be  to  hold 
100  bushels  ? 

9.  Find  the  lateral  surface  of  an  equilateral  triangular 
pyramid,  the  perimeter  of  the  base  being  12  m.  and  the  slant 
height  14  m. 

10.  Find  the  volume  of  a  square  pyramid,  the  perimeter 
of  whose  base  is  16  ft.  and  whose  altitude  is  9  ft. 

11.  What  is  the  volume  of  the  largest  cone  that  can  be 
cut  from  a  pyramid  whose  base  is  6  feet  square,  and  whose 
slant  height  is  15  feet  ? 

]  12.    A  cylindrical  tank  is  14  ft.  deep  and  6  feet  in  diam- 

eter.   Find  the  cost  of  cementing  the  sides  at  90  J?  a  sq.  yard. 

13.  Find  the  capacity  in  gallons  of  a  cylindrical  cistern 
whose  inside  diameter  is  6  feet,  and  whose  depth  is  7  feet. 

14.  Find  the  capacity  in  Kl.  of  a  cylindrical  cistern  whose 
inside  diameter  is  4  m.,  and  whose  altitude  is  5  m. 


APPENDIX. 


MARINERS'  MEASURES. 

532.  TABLE. 

6  feet  =  1  fathom 

120  fathoms  =  1  cable-length 

7J  cable-lengths     =  1  mile 

1.15  statute  miles  =  1  nautical  mile 
3  nautical  miles     =  1  marine  league 

OTHER   LINEAR   MEASURES   (APPROXIMATE). 

4  inches  =  1  hand  3.3  feet  =  1  pace 

9  inches  =  1  span  5  paces  =  1  rod 

SURVEYORS'  LINEAR   MEASURE. 

533.  The  Surveyors'  Chain  is  made  of  100  links,  each  link 
being  7.92  inches  long.  It  is  called  Gunter's  Chain,  from  the 
name  of  the  inventor. 

The  steel  measuring  tape  is  100  feet  long,  each  foot  being 
divided  into  tenths  and  hundredths. 

397 


398 


APPENDIX. 


534. 


SURVEYORS'   SQUARE  MEASURE. 

TABLE. 

625  sq.  links     =  1  sq.  rod  .  .  .  sq.  rd. 
16  sq.  rods        =  1  sq.  chain  .  .  .  sq.  ch. 
10  sq.  chains  1 

or  I"  ~  ^  ^^^®  ...  A. 

160  sq.  rods   J 

640  acres  =  1  sq.  mile  .  .  .  sq.  mi. 

1  sq.  mi.  =  640  A.  =  6400  sq.  ch.  ==  102,400  sq.  rd. 
=  64,000,000  sq.  1. 


GOVERNMENT  LANDS. 

535.  The  government  lands  of  the  United  States  are 
divided  by  parallels  and  meridians  into  Townships,  6  miles 

square.  Each  township 
is  divided  into  36  square 
miles,  or  Sections.  Each 
section  is  subdivided  into 
half-sections  and  quarter- 
sections. 

In  surveying  the  pub- 
lic lands,  lines  6  miles 
apart  are  run  from  east 
to  west  and  from  north 
to  south,  dividing  the  ter- 
ritory into  square  town- 
ships. 

An  east  and  west  line  is  established  as  a  Base  Line,  and 
a  north  and  south  line  as  a  Principal  Meridian. 

A  line  of  townships  running  east  and  west  is  called  a 
Tier,  and  a  line  of  townships  running  north  and  south  is 
called  a  Range. 

Any  township  is  designated  by  its  number  north  or  south 


1 — 1 — 1 — 

TOWNSHIP 

5 

NORTH 

TOWNi 

5HIP 

4 

NOF 

TH 

3J 

33 

:a 

3J 

30 

^ 

73 

33 

33 

33 

->- 

->- 

->- 

->— 

->— 

^^H 

— j>— 

->— 

->- 

->— ■ 

7 

z 

z 

7 

7 

— > 

z 

7 

^ 

Z 

O 

o 

o 

O 

(7) 

Uz 

o 

O 

C) 

o 

-u\- 

-m- 

-m- 

-m- 

-m- 

->ili 

— m— 

-m- 

-rn- 

-m-- 

01 

■*>. 

co 

h3 

■" 

I-- 

ro 

co 

■t^ 

Ol 

BA 

SE 

i 

LIT 

ME 

m 
-co- 

■i- 

-%- 

-%- 

2 
r-j 

* 

> 

m 

m 
> 

TO 

WN5 

HIP 

4 

> 

SOI 

TH 

1         1 
TOWNSHIP 

5 

SOUTH 

APPENDIX. 


399 


of  the  base  line,  and  its  number  east  or  west  from  the 
principal  meridian. 

Thus,  a  township  that  is  in  the  15th  tier  north  of  the  base 
line,  and  in  the  28th  range  east  of  the  4th  principal  meridian, 
is  designated : 

T.15K  I1.28E.  4th  P.M. 

There  being  36  sections  in  a  township,  each  section  is 
designated  by  a  number. 

The  numbering  begins  at  the  N.E.  corner,  increasing 
toward  the  west  and  east,  as  shown  in  the  accompanying 
diagram. 


TOWNSHIP 

N 


SECTION 

N 
ONE  MILE 


W 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

X 


Ul 

-1 

330  A. 

N.E.^ 
160  A. 

Ul 

I 

of 

S.E.M 
80  A. 

ofS.E.« 
40  A. 

of8.E.Ji 

40  A, 

ONE  MILE 
S 


Each  section  is  divided  into  4  quarter-sections,  containing 
160  acres  each,  and  named,  as  shown  in  the  diagram. 


MISCELLANEOUS  MEASURES   OF   WEIGHT. 

636.  AVOIRDUPOIS  WEIGHT. 

14  lb.  =  1  Stone 

56  lb.  Butter  =  1  Firkin 

100  lb.  Grain  =  1  Cental 

100  lb.  Dried  Fish  =  1  Quintal 

100  lb.  Nails  =  1  Keg 

196  lb.  Flour  =  1  Barrel 

200  lb.  Beef  or  Pork  =  1  Barrel 
280  lb.  Salt  at  N.  Y.  Works  =  1  Barrel 


400 


APPENDIX. 


Grain,  vegetables,  seeds,  coal,  etc.,  are  sold  by  the  bushel. 
Grain,  seeds,  and  very  small  fruits  are  sold  by  stricken 
measure.  Large  fruits,  vegetables,  corn  in  the  ear,  etc.,  are 
sold  by  heaped  measure.  The  measure  should  be  heaped  or 
rounded  as  high  as  6  inches  above  the  top  of  the  measure. 

The  standard  unit  for  the  United  States  is  the  "Winchester 
Bushel.  It  is  cylindrical  in  form,  18^  inches  in  diameter, 
and  8  inches  deep.     It  contains  2150.42  cubic  inches. 

It  is  customary  in  estimating  the  number  of  bushels 
that  will  be  contained  in  a  given  bin  or  space,  to  consider 
the  bushel  as  occupying  1\  cu.  ft.  of  space,  nearly.  As  there 
are  1728  cu.  in.  in  1  cu.  ft.  there  must  be  as  many  cubic  feet 
in  a  bushel,  as  1728  is  contained  times  in  2150.42,  or  1\, 
nearly. 

537.  The  following  table  shows  the  number  of  pounds  in 
a  legal  bushel,  of  different  commodities,  in  various  states : 


Wheat 

Indian  Corn,  shelled 

Oats 

Barley 

Buckwheat     .     .     . 

Eye 

Clover  Seed  .  .  . 
Timothy  Seed  .  . 
Blue  Grass  Seed 


O 

2; 

-3 

H 

Q 

i 

0 

^ 

O 

60 
50 
35 
48 
52 
56 
60 
45 
14 

>• 
M 

60 

56 

33i 

48 

52 

56 

60 

45 

14 

■i 
J 

60 
56 
32 
32 

32 

i 

< 

60 
56 
30 
46 
46 
56 

o 

60 
56 
32 
48 
42 
56 
60 

■jr. 

60 
56 
82 
48 
42 
56 
60 

6 

60 
52 
35 
48 
52 
56 
60 
45 
14 

60 
56 
30 
48 
50 
56 
64 

^ 

60 

58 
32 
48 
48 
56 
60 
45 

60 
54 

48 
50 

6 
O 

60 
56 
32 

48 

56 
60 

O 

60 
56 
34 
46 
42 
56 
60 

ii 

z 

60 
56 
32 

47 
48 
56 

> 

60 
56 
82 
46 
46 
56 

< 

60 
56 
36 
45 
42 
56 
60 

22 

60 
52 
32 

50 
40 
54 

56 
56 

28 

45 
56 

60 
56 

60 
52 
32 
48 
40 
54 
60 
45 
14 

60 
56 
32 
48 
50 
56 
60 
45 
14 

60 
56 
32 
48 
42 
56 
60 
46 

Beans,  peas,  and  potatoes  usually  60  lb. ;  in  N.  Y.,  beans 
62  1b. 

Coal,  80  lb.,  except  Ind.,  70  or  80,  and  Ky.  76  lb. 

Salt:  111.,  50  lb.  common,  or  55  lb.  fine, 

K.  J.,  56  lb.,  Ind.,  Ky.,  and  Iowa  50  lb., 

Penn,  80  lb.  coarse,  70  lb.  ground,  or  62  lb.  fine. 


APPENDIX.  401 

APOTHECARIES'  FLUID  MEASURE. 

538.  APPROXIMATES. 

1  fluid  drachm  =  45  drops  of  water,  or  a  common  tea- 

spoonful 
1  fluid  ounce  =  2  tablespoonfuls 

4  fluid  ounces  =  1  gill,  or  1  small  teacupful 

4-  gill  =  4  tablespoonfuls,  or  a  wine-glass 

1  pint  of  pure  water  =  1  pound 
4  teaspoonfuls  '  =  1  tablespoonful 

FARMERS'   ESTIMATES. 

539.  To  find  the  number  of  bushels  in  a  bin  or  granary, 
Divide  the  number  of  cubic  feet  in  the  bin  or  granary  by  1\. 

To  find  how  large  a  bin  will  contain  a  given  number  of 
bushels, 

Multiply  the  number  of  bushels  by  1^. 

The  result  is  the  number  of  cubic  feet  in  the  required  bin. 

To  find  the  number  of  gallons  of  water  in  a  cistern  or  tank, 

Multiply  the  member  of  cubic  feet  in  the  cistern  or  tank 
by  7|. 

To  find  how  large  a  cistern  will  hold  a  given  number  of 
gallons, 

Divide  the  number  of  gallons  by  7|-. 

The  result  will  be  the  number  of  cubic  feet  in  the  required 
cistern. 

To  find  how  many  bushels  of  shelled  com  in  a  given  number 
of  bushels  of  corn  in  the  ear, 

Divide  the  number  of  bushels  of  corn  in  the  ear  by  2. 


402  ,  APPENDIX. 

To  find  the  number  of  tons  of  hay  in  a  mow  or  stack, 

Divide  the  number  of  cubic  feet  in  the  mow  or  stack  by  500. 
If  clover,  divide  by  550. 

Note.  — Hay  should  be  well  pressed  down. 

It  is  estimated  that  horses  and  sheep  consume  daily  about 
3  pounds  of  hay  for  each  100  pounds  of  weight.  Cows  and 
oxen  about  2^  pounds.  As  food  for  stock,  corn  and  oats 
are  equivalent  to  about  twice  their  weight  in  hay.  Cotton- 
seed meal  about  3  times  its  weight  in  hay. 

Net  weight  of  fat  beeves  is  about  f  of  live  weight,  of  fat 
hogs,  I ;  of  fat  sheep,  J. 

BUSINESS   FORMS. 

540.  A  written  acknowledgment  that  money,  or  its  equiva- 
lent value,  has  been  received,  is  called  a  Receipt. 

Receipt  on  Account. 
$325y\^^  Chicago,  111.,  Dec.  1,  1900. 

Keceived  from  W.  S.  Smyth,  three  hundred  twenty-five 

dollars  on  account.  a     t\   td 

A.  D.  i'erkins. 

Receipt  in  Full. 

Detroit,  Mich.,  July  8,  1901. 
Eeceived  from  Charles  Anderson,  thirty -seven  and  -fi^ 
dollars,  in  full  of  account. 

$37tV7  H.  King  &  Co. 

Receipt  for  Rent. 
.f  97_Y^  Cincinnati,  O.,  Jan.  6,  1900. 

Received  from  H.  K.  Pierce,  ninety-seven  dollars  for  rent 
of  dwelling  No.  504  McBride  St.,  from  July  15  to  October 

15,  1900.  rj  r^ 

'  Henry  Quilton. 


APPENDIX.  403 

Order  for  Money. 

San  Francisco,  Cal.,  March  10,  1899. 
Messrs.  David  Hunter  &  Sons, 
Little  Rock,  Ark. 

Please  pay  to  bearer,  Mr.  Jacob  Schmidt,  one  hundred 
fifteen  dollars,  and  charge  to  the  account  of 

Nicholas  Grumbach. 

Order  for  Goods. 

TopEKA,  Kans.,  May  30,  1899. 
Messrs.  J.  H.  Andrews  &  Co., 
Denver,  Col. 

Please  deliver  to  George  M.  White  goods  to  the  value  of 
one  hundred  thirty-two  dollars,  and  charge  the  same  to  my 
account. 

Jacob  Riis. 

A  Check  is  a  written  order,  addressed  to  a  bank  by  a 
depositor,  requesting  the  payment  of  a  certain  amount  of 
money  to  a  person  named,  or  to  that  person's  order. 

Bank  Check. 

Bank  of  Commercb 

Onno.    humAnoA.     5^1/ytogm.    nrr\A -Mr  X> ollars 

liy^rr^.  IT  SoiinAnrx. 


404 


APPENDIX. 


COMPUTING    TAXES. 

541.  A  tax  list  may  be  extended,  with  a  minimum  liability 
to  make  errors,  by  arranging  a  table  showing  the  tax  of 
units,  tens,  hundreds,  thousands,  etc.,  of  dollars.  The  rate 
used  in  the  following  is  ^Yiis  mills  on  a  dollar. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$.00523 

$10 

$0.0523 

$100 

$0,523 

$1000 

$  5.23 

2 

.01046 

20 

0.1046 

200 

1.046 

2000 

10.46 

3 

.01569 

30 

0.1569 

300 

1.569 

3000 

15.69 

4 

.02092 

40 

0.2092 

400 

2.092 

4000 

20.92 

5 

.02615 

50 

0.2615 

500 

2.615 

5000 

26.15 

6 

.03138 

60 

0.3138 

600 

3.138 

6000 

31.38 

7 

.03661 

70 

0.3661 

700 

3.661 

7000 

36.61 

8 

.04184 

80 

0.4184 

800 

4.184 

8000 

41.84 

9 

.04707 

90 

0.4707 

900 

4.707 

9000 

47.07 

Make  the  table  by  finding  the  tax  on  units  of  dollars  to 
$9.  The  tax  on  the  tens,  hundreds,  etc.,  of  dollars  is  found 
by  removing  the  decimal  point  one  or  more  places  to  the 
right. 

What  is  the  tax  at  the  above  rate  on  ^  7856  ? 


Solution. 

Tax  on  $7000,  as  per  table,  =$36.61 
Tax  on  $800,  as  per  table,     =      4.184 
Tax  on  $50,  as  per  table,       =        .2615 
Tax  on  $6,  as  per  table,        =        .0313 


Total  tax  on  $7856,         $41.0868 

The  same  result  may  be  found  by  multiplying  $7856  by 
the  rate. 


APPENDIX. 


405 


542.   The  following  table  shows  the  legal  rate  of  interest 
in  each  of  the  states  and  territories  in  the  United  States. 


TABLE. 


State. 

Rate. 

State. 

Rate. 

Alabama 

S 

8 

Montana 

8 

Any, 

Arizona 

7 

Any. 

Nebraska 

7 

10 

Arkansas   ....... 

6 

10 

Nevada 

7 

Any. 

California 

T 

Any. 

New  Hampshire    .... 

6 

6 

Colorado 

8 

Any. 

New  Jersey 

6 

6 

Connecticut 

6 

6 

New  Mexico 

6 

12 

Delaware 

6 

6 

New  York 

6 

6 

District  of  Columbia    .     .     . 

6 

10 

North  Carolina      .... 

6 

6 

Florida 

8 

10 

North  Dakota 

7 

12 

Georgia 

T 

8 

Ohio 

6 

8 

Idaho     

7 

12 

Oklahoma 

7 

12 

Illinois 

5 

7 

Oregon 

6 

10 

Indiana 

6 

8 

Pennsylvania 

6 

6 

Indian  Territory      .... 

6 

10 

Rhode  Island 

6 

Anv. 

Iowa 

6 

8 

South  Carolina 

7 

8 

Kansas 

6 

10 

South  Dakota 

7 

12 

Kentucky 

6 

6 

Tennessee 

6 

6 

Louisiana  ....... 

5 

8 

Texas 

6 

10 

Maine 

6 

A.J. 

Utah 

8 

Any. 

Maryland 

6 

Vermont 

6 

6 

Massachusetts 

« 

Any. 

Virginia 

6 

6 

Michigan 

5 

7 

Washington 

6 

12 

Minnesota 

6 

10 

West  Virginia 

6 

6 

Mississippi 

6 

10 

Wisconsin 

6 

10 

Missouri 

6 

8 

Wyoming 

8 

12 

When  no  rate  is  mentioned  in  a  note  or  other  contract,  the  rate  in  the  left-hand 
column  may  be  collected  by  law.  Any  rate  not  exceeding  that  in  the  right-hand  column 
may  be  collected  when  specified  in  the  note  or  contract. 


EXCHANGE. 

543.  Exchange  is  a  method  of  making  payments  between 
distant  places  without  transmitting  the  money. 

If  A  of  Boston  owes  f  1000  to  X  of  Denver,  and  B  of 
Denver  owes  $  1000  to  Y  of  Boston,  B  of  Denver  may  pay 
X  of  Denver  for  an  order  on  A  of  Boston  to  pay  the  $  1000 
to  Y  of  Boston.  A,  therefore,  may  send  this  order  to  Y 
who  presents  it  to  A  and  receives  his  money.  Thus,  the 
two  debts  are  discharged  without  the  sending  of  money. 

Such  transactions  are  usually  carried  on  through  banks, 
which  charge  a  small  fee  for  their  services. 


406  APPENDIX. 

544.  The  written  order,  directing  one  person  to  pay  a 
certain  sum  to  another,  is  called  a  Draft  or  Bill  of  Exchange. 

545.  The  signer  of  a  draft  is  called  the  Drawer,  the  person 
to  whom  it  is  addressed  is  the  Drawee,  and  the  person  to 
whom  it  is  payable  is  the  Payee. 

546.  A  draft  payable  on  presentation  to  the  drawee  is  a 
Sight  Draft.  A  draft  payable  at  a  specified  time  after  pres- 
entation is  a  Time  Draft. 

TIME    DRAFT. 

^  foc]^  SanTranci&co,  Oct..  u.   1 8 9_q_ 

At.  torn. .   AmiW  9,1/:^  t.    r>ny  to  tke 

order  nf   l?n^)^nt.~T}\jrrrr\n/i  SniiAonrrrxnm. 

Ot\p.  hxirrxAnsA,,  Txx/rKo.  a/nd.  ipo    Do  liars 

yalue  received,  and  charge  tfic  sameioihe  account  of 

To   hf.C  ZmA/iAK 

vr  o  -+-    o    n  ■  A  A  IT,  Kn/rxQ.'Y  Co. 

Three  days  of  grace  are  usually  allowed  on  time  drafts. 

547.  If  the  drawee  accepts  a  draft,  he  writes  the  word 
"Accepted"  across  its  face,  and  signs  his  name,  with  the 
date  of  acceptance.  This  is  an  agreement  to  pay  it,  and  is 
called  the  **  Acceptance  "  of  the  draft. 

DOMESTIC  EXCHANGE. 

548.  Exchange  between  places  that  are  in  the  same  country 
is  called  Domestic  Exchange. 

Note.  —  When  a  draft  sells  for  its  face,  it  is  at  par  ;  when  for  less 
than  its  face,  it  is  at  a  discount ;  when  for  more  than  its  face,  it  is  at  a 
premium. 


APPENDIX.  407 

1.  What  will  be  the  cost  of  a  sight  draft  for  $  500  at  J% 

premium  ? 

Solution.  —  Since    exchange    is 
$  100  +  $  .0025  =  $1.0025      ^^  ^  premium,  each  dollar  of  the 

draft  will  cost  $  1.0025.     Therefore 
$  1.0025  X  500  =  $  501.25      ^  $  500   draft   will  cost   500  times 

$  1.0025. 

2.  What  is  the  cost  of  a  New  York  draft  of  $  1000,  at 
1%  discount? 

^  -j^  QQ ^  Q^_.^  99  Solution.  — Since  exchange  is  at  a  dis- 

count, each  dollar  of  the  draft  will  cost 
$  .99  X 1000  =  $  990       ^  99^  and  $  1000  will  cost  1000  times  |  .99. 

3.  What  is  the  cost  of  a  sight  draft  on  Denver  for  $  5000 
at  H%  premium? 

Find  the  cost  of  the  following  sight  drafts. 

4.  On  New  Orleans  at  i%  discount  for  $  498. 

5.  On  Cincinnati  at  lj%  premium  for  $  3000. 

6.  On  Boston  at  i%  premium  for  $  875. 

7.  On  Buffalo  at  i%  discount  for  $  750. 

8.  What  is  the  cost  in  Chicago  of  a  draft  on  New  York 
for  $  1000  payable  2  months  after  sight,  at  |%  premium  ? 

Solution.  —  At  |  %  premium,  each  dollar  of  the  draft  will  cost  at 
sight  $1,005.      But  since  the  draft  is  not  payable  until  2  mo.  3  da. 

after  sight,  the  banker  allows 

$  1  +  $  .005  =  $  1.005       ^he  bank  discount  at  the  legal 

$  1.005  —  $  .0105  =  $  .9945        rate  of   interest   in  Illinois  for 

$  .9945  X  1000  =  $  994.50     ^^'  ^  "'^^  ^  ^^-    ^^^  ^'^^^^^ 

amounts  to  |  .0105  on  each  dol- 
lar. Subtracting  this  discount  from  $  1.005,  we  have  the  cost  of  f  1  of 
the  draft,  or  $.9945.  A  draft  of  $1000  will  cost $1000  times  .9945, 
or  $994.50. 

9.  What  will  be  the  cost  in  New  York  of  a  Denver 
draft  for  f  1000  payable  1  month  after  sight,  exchange 
being  at  1%  discount. 


408  APPENDIX. 

10.  Find  the  cost  in  Savannah,  G-a.,  of  a  draft  on  Phila- 
delphia for  $5000,  at  60  days'  sight,  at  1%  premium,  and 
interest  at  6%. 

11.  What  is  the  cost  in  St.  Louis  of  a  $500  draft  on 
New  York  at  30  days'  sight,  at  1^%  premium,  and  interest 
at  6%  ? 

12.  How  large  a  sight  draft  on  Chicago  can  be  bought  for 
$3030,  exchange  being  at  1%  premium? 

Solution.  —  Since  $  1  of  the  draft 

$1  +  $.01  =  $1.01       win   cost  $1.01,  as  many  dollars  can 

$3030 -J- $1.01  =  $3000      be  purchased  for  $3030,  as  $1.01  is 

contained  times  in  $3030,  or  $3000. 

13.  What  was  the  face  of  a  sight  draft  on  Boston  pur- 
chased for  $1015,  exchange  being  at  1J%  premium  ? 

14.  How  large  a  sight  draft  on  Nashville  can  be  pur- 
chased for  $2550,  when  the  exchange  is  at  J  %  discount? 

15.  How  large  a  draft  on  Rochester  can  be  purchased  in 
Boston  for  $5000  at  60  days'  sight,  the  premium  being  1^%, 
and  interest  6%? 

$  1  +  $  .015  =  $  1.015  Solution.  —At  li%  premium, 

$  1.015  —  $  .0105  =  $  1.0045        $  1  of  the  draft  will  cost  $  1.015 
$  5000  --  $  1.0045  =  $  4977.60  +  ^^  **^^^" 

But  since  it  is  not  payable  till  63  days  after  sight,  the  Boston 
banker  will  allow  6  %  discount  for  that  time,  which  is  $  .0105  on  each 
dollar.  Deducting  this  from  the  sight  price  of  $  1,  we  have  $  1.0045, 
the  cost  of  $1  of  the  draft.  Therefore,  since  $1.0045  will  purchase 
$  1  of  the  draft,  $  5000  will  purchase  as  many  dollars  as  $  1.0045  is 
contained  times  in  $5000,  or  $4977.60  +  . 

16.  How  large  a  draft  can  be  purchased  for  $2500,  60 
days  after  sight  at  1%  discount,  when  interest  is  6%  ? 

17.  Find  the  face  of  a  draft  on  Charleston  at  90  days' 
sight,  that  can  be  purchased  in  New  York  for  $3500, 
exchange  being  at  1%  premium,  and  interest  6%. 


APPENDIX.  409 

18.  What  is  the  face  of  a  draft  on  New  York  at  90  days' 
sight,  which  may  be  purchased  for  $2000,  exchange  being 
\<^o  discount,  and  interest  5%  ? 

FOREIGN  EXCHANGE. 

549.  Exchange  between  places  in  different  countries  is 
Foreign  Exchange. 

Drafts  drawn  on  foreign  countries  are  expressed  in  the 
money  of  the  country  in  which  they  are  payable. 

550.  Foreign  bills  of  exchange  are  usually  made  in  sets 
of  three,  of  the  same  date  and  tenor,  and  named  ^?'s^,  second, 
and  tliird  of  exchange.  These  are  sent  by  different  mails 
or  routes.  When  either  of  the  three  is  paid,  the  other  two 
are  void. 

551.  Exchange  in  European  countries  is  done  chiefly 
through  large  commercial  cities,  as  London,  Paris,  Hamburg, 
Amsterdam,  etc. 

552.  Bills  drawn  on  England,  Ireland,  or  Scotland  are 
called  Sterling  Bills,  and  the  current  value  of  a  Pound  Ster- 
ling is  quoted  in  United  States  money. 

The  foreign  exchange  of  the  United  States  is  done  chiefly 
with  Great  Britain,  France,  and  Germany. 

Note. — The  Secretary  of  the  Treasury  of  this  country  publishes 
annually  the  values  of  all  foreign  currency  in  United  States  money. 

553.  The  English  pound  sterling  or  sovereign  is  valued  at 
$4.8665  in  United  States  gold. 

The  French  franc  is  valued  at  $  .193,  or  5.18  francs  for  $  1. 

The  German  mark  is  valued  at  $.238  in  United  States 
money,  or  4  marks  for  $  .952. 

The  above  are  the  values  when  exchange  is  at  par.  Their 
current  or  commercial  values  may  be  above  or  below  par. 


410  APPENDIX. 

1.  What  is  the  cost  in  New  York  of  a  sight  draft  on 
Liverpool  for  £520  12s.  6d.  when  exchange  is  $4,875  to 
the  pound  sterling  ? 

Solution.  —£520  12s.  6d.  =  £520.625. 

Since  1  pound  is  wortli  $4,875,  £520.625  are  worth  520.625  times 
$4,875,  or  $2538.046+,  the  cost  of  the  draft. 

2.  What  is  the  cost  of  a  bill  on  Paris  for  625  francs, 
exchange  being  at  5.20  francs  to  the  dollar  ? 

Solution.  — Since  5.20  francs  cost  $  1,  625  francs  will  cost  as  many- 
dollars  as  5.20  francs  is  contained  times  in  625  francs,  or  $120,192+, 
the  cost  of  the  draft. 

3.  Find  the  cost  of  a  bill  on  Glasgow  for  £384  15s.  9d., 
exciiange  being  $  4.87  for  a  pound. 

4.  What  is  the  cost  of  a  draft  on  Berlin  for  1250  marks, 
exchange  being  at  $  .95  for  4  marks. 

Solution.  — Since  4  marks  are  worth  $.95,  1  mark  is  worth  ^  of 
$.95,  and  1250  marks  are  worth  1250  times  |  of  $.95,  or  $296,875, 
the  cost. 

5.  How  much  must  be  paid  for  a  draft  on  Frankfort  for 
648  marks,  exchange  being  $  .95^  per  4  marks  ? 

6.  How  much  must  be  paid  for  a  bill  of  exchange 
on  Havre  1274.28  francs,  exchange  being  5.18  francs  to  the 
dollar  ? 

7.  What  is  the  face  of  a  bill  of  exchange  at  sight  on 
London,  purchased  in  New  York  for  $  3500,  exchange  being 
$4.86  for  a  pound  sterling  ? 

Solution.  —  Since  £  1  is  purchased  for  $  4.86,  as  many  pounds  can 
be  purchased  for  $3500  as  $4.86  is  contained  times  in  $3500,  or 
£720  3s.  3d,  the  face. 

8.  How  large  a  bill  on  Paris  can  be  purchased  for 
$2500,  exchange  being  5.15  francs  to  a  dollar? 

9.  What  is  the  face  of  a  bill  on  Dublin  which  costs 
$4865  in  United  States  gold,  exchange  at  $4,865? 

10.  How  large  a  bill  on  Hamburg  can  be  purchased  for 
$  3810,  exchange  95|  ? 


ANSWERS    TO    EXAMPLES    IN    HEATH'S 
COMPLETE   PRACTICAL  ARITHMETIC. 


Article  47. 

1. 

6116. 

16. 

$235,222. 

31. 

$10,290.28. 

46. 

99,820 

2. 

8915. 

17. 

$148,285. 

32. 

1364. 

votes. 

3. 

18,441. 

18. 

28,007. 

33. 

$240.75. 

47. 

$66,440. 

4. 

21,365. 

19. 

125,448. 

34. 

$290.42. 

48. 

7,793,300 

5. 

$771.40. 

20. 

4492. 

35. 

$134.90. 

sq.  mi. 

6. 

$1287.873. 

21. 

1,197,972. 

36. 

$1276. 

49. 

33,897  ft. 

7. 

$821,191. 

22. 

$48,978. 

37. 

$8944. 

50. 

11,783  lb. 

8. 

$5717.189. 

23. 

$131,165. 

39. 

$31,843. 

51. 

4996. 

9. 

68,605. 

24. 

42,681. 

40. 

$59,584 

52. 

39,363. 

10. 

62,054. 

25. 

58,593. 

41. 

541  mi. 

53. 

$  208.43. 

11. 

60,088. 

26. 

463,090. 

42. 

1861. 

54. 

$316,639. 

12. 

2,953,660. 

27. 

491,467. 

43. 

5,892,906. 

55. 

429,879. 

13. 

2,760,578. 

28. 

3122. 

44. 

$221.50. 

56. 

944,835. 

14. 

2,651,844. 

29. 

890,407„ 

45. 

1693  lb. 

57. 

$1534.028. 

15. 

$197.37. 

30. 

103,019. 

Article  56. 

2. 

276. 

13. 

6927. 

24. 

1036. 

35. 

$22.90. 

3. 

137. 

14. 

3698. 

25. 

964. 

36. 

$  .416. 

4. 

19. 

15. 

6499. 

26. 

3979. 

37. 

$.598. 

5. 

25. 

16. 

1273. 

27. 

1019. 

38. 

$11,361. 

6. 

66. 

17. 

156. 

28. 

585. 

39. 

$13,533. 

7. 

69. 

18. 

1474. 

29. 

297. 

40. 

3269. 

8. 

618. 

19. 

2997. 

30. 

16,677. 

42. 

1128. 

9. 

162. 

20. 

1338. 

31. 

4315. 

43. 

32,376. 

10. 

67. 

21. 

7082. 

32. 

10,388. 

44. 

31,676. 

li. 

219. 

22. 

1482. 

33. 

1450. 

45. 

16,668. 

12c 

2146. 

23. 

996. 

34. 
411 

12,223. 

46. 

4025. 

41! 

2 

ANSWERS. 

47. 

16,042. 

63. 

$4126.832. 

59. 

11,369. 

65. 

$149.36. 

48. 

$.129. 

54. 

7529. 

60. 

$25,967. 

66. 

2462. 

49. 

$10,449. 

55. 

25,350. 

61. 

41,276. 

67. 

1925. 

50. 

$1,244. 

56. 

21,996. 

62. 

$813,875. 

68. 

$a725. 

51. 

$8,936. 

57. 

6039. 

63. 

$2.33. 

52. 

$19.51. 

58. 

7724. 

64. 

166  A. 

Article  56. 

• 

69. 

284  yr. ;  42 

yr. 

75.    483  mi 

81. 

$5250. 

70. 

$6875. 

76.   6855  sq.  mi. 

82. 

$2131. 

71. 

489  boys. 

77.    13,108  votes. 

83. 

$1275. 

72. 

13,289,220. 

78.    1,763,100. 

84. 

841  yd. 

73. 

12,789  ft. 

79.    175  mi 

85. 

3345. 

74. 

$1,226,274,478. 

80.    $28,500. 

Article  65. 

27. 

648. 

41. 

23,000. 

55. 

$103.88. 

69. 

31,680  ft. 

28. 

2065. 

42. 

53,883. 

56. 

$98,615. 

70. 

$9.88. 

29. 

2912. 

43. 

703,676. 

57. 

$164,565, 

71. 

$101.08. 

30. 

1698. 

44. 

5,538,972. 

58. 

$89,948. 

72. 

7040. 

31. 

2982. 

45. 

785,928. 

59. 

$  5.625. 

73. 

$  18,624. 

32. 

4232. 

46. 

1,489,686. 

60. 

$  18.66. 

74. 

26,880  lb. 

33. 

4428. 

47. 

4,384,740. 

61. 

$11,488. 

75. 

1344  hr. 

34. 

2630. 

48. 

5,838,760. 

62. 

$29.04. 

76. 

896  cu.  ft. 

35. 

43,130. 

49. 

2,558,070. 

63. 

$13.75. 

77. 

$151.50. 

36. 

34,008. 

50. 

1,111,104. 

64. 

$114.75. 

78. 

$75. 

37. 

7580. 

51. 

$5.04. 

65. 

$12.96. 

79. 

320  qt. 

38. 

11,530. 

52. 

$14.35. 

66. 

1584  pens 

I. 

80. 

864  sq.  in. 

39. 

35,232. 

53. 

$23.64. 

67. 

3360  bu. 

81. 

23,688  lb. 

40. 

34,562. 

54. 

$61.25. 

68. 

22,464. 

82. 

1296  units. 

Article  67. 

1. 

1250. 

9.    $395. 

18. 

8580. 

2. 

$36.40. 

10.    $469. 

19. 

20,160. 

3. 

5080. 

11.    $360. 

20. 

19,750. 

4. 

3090. 

12.   240,000. 

21. 

78, 

,750. 

5. 

7860. 

13.    $293,000. 

22. 

97,200. 

6. 

2800. 

14.    439,800,000 

23. 

131,200. 

7. 

3600. 

15.    2,873,200,000. 

24. 

356,400. 

8. 

28,400. 

17.   7360. 

25. 

278,600. 

ANSWERS.  413 

26.  386,100.  55.  268,830.  83.  239,112;  116,268. 

27.  1,107,000.  56.  341,478,000.  84.  196,840  ;  308,416. 

28.  2,130,000.  57.  342,954.  85.  $19,656. 

29.  35,880,000.  58.  28,460,432.  86.  28,160  yd. 

30.  259,200,000.  60.  68,376.  87.  6290  bu. 

31.  8,874,000,000.  61.  151,032.  88.  110,880  ft. 

35.  89,148.  62.  196,564.  89.  324,000  1b. 

36.  408,498.  63.  180,438.  90.  ^1980. 

37.  127,050.  64.  199,681.  91.  196,0001b. 

38.  $1701.56.  65.  160,947.  92.  1512  mi. 

39.  $7615.11.  66.  331,224.  93.  $33,760. 

40.  5,678,986.  67.  347,512.  94.  29,344  1b. 

41.  $123,970.  69.  61,184.  95.  $661.50. 

42.  4,992,232.  70.  35,424.  96.  $21,000. 

43.  162,582.  71.  21,588.  97.  $2080. 

44.  65,492,908.  72.  42,133.  98.  4,498,660. 

45.  224,016.  73.  41,956.  99.  8760  hr. 

46.  42,739,736.  74.  32,186.  100.  36,266,000  A. 

47.  $1525.68.  75.  48,885.  101.  $1,043,619. 

48.  350,649,186.  76.  46,064.  102.  $612.11. 

49.  122,688.  77.  $4914.  103.  $1043.75. 

50.  93,788,068.  78.  $38.10.  104.  10,878  da. 

51.  71,800.  79.  7446  mi.  105.  116  mi. 

52.  154,629,780.  80.  $203.40.  106.  200  mi. 

53.  270,060.  81.  $301,224.  107.  6,043,500  1b. 

54.  418,460,000.  82.  $98,022.  108.  $180.60;  $33.60. 


Article  82. 

6.  511.  17.  14,960.  28.  $30.296r\.  38.  $35.45. 

7.  5541.  18.  6179.  29.  $24.40.  39.  41,097;  3. 

8.  1172.  19.  11,669.  30.  $73.10.  40.  $9962; 

9.  901.  20.  17,460|.  31.  $31.88.  $7969§. 

10.  412f.  21.  4976|.  32.  $30,303.  41.  4315|. 

11.  796^.  22.  10,711f.  33.  62|.  42.  $300. 

12.  349|.  23.  10,940^.  34.  507  bbls.  43.  48  miles  an 

13.  1293.  24.  $120,001.  35.  $5.25.  hour. 

14.  331^.  25.  $24,001.  36.  $12.08.  44.  $5.10. 

15.  6882.  26.  $30,001.  37.  $13.64.  45.  775  bu. 

16.  4049.  27.  $90,001. 

46.    Rose,  $15.09;  James,  $60.36.          47.    $.96.  48.   965. 


4V 

1: 

ANSWERS. 

Article  83. 

7. 

384.92. 

10. 

387. 

13. 

28.006. 

16. 

$2.89. 

8. 

296.48. 

11. 

28. 

14. 

198.751. 

17. 

$1,398. 

9. 

169. 

12. 

39.642. 

15. 

$  13.805. 

18. 

$29.84. 

Article  85. 

2. 

161^. 

18. 

2576|f. 

34. 

260,985Hff. 

50. 

23  da. 

3. 

807^V- 

19. 

7113||. 

35. 

72,632^110. 

51. 

$127.    ' 

4. 

604. 

20. 

13,085/3. 

36. 

7445^W^. 

52. 

54  cu.  ft. 

6. 

322^1. 

21. 

38,039if. 

37. 

20,331}|H- 

53. 

r  The  latter. 
I  $.50. 

6. 

380^^^. 

22. 

4406|f 

38. 

9465i|ff. 

7. 

483f^. 

23. 

$6.04. 

39. 

849  carriages, 

,  54. 

45  horses. 

8. 

413|f. 

24. 

$7.63. 

40. 

990IH. 

55. 

f  54  mi.  per 
1     hour. 

9. 

779if. 

25. 

$  .849. 

41. 

$968. 

10. 

686||. 

26. 

$2,125. 

42. 

129  pk. 

56. 

15  yr. 

11. 

472H. 

27. 

$6848.76|f. 

43. 

752  bu. 

57. 

318. 

12. 

573ff. 

28. 

$8235.11. 

44. 

42  bbl. 

58. 

14  da. 

13. 

69H|. 

29. 

$7511.23. 

45. 

32  bu. 

59. 

$156.25. 

14. 

3960^3^. 

30. 

$2605.102. 

46. 

36  hr. 

60. 

$6.50. 

15. 

778H. 

31. 

28,166^\%%. 

47. 

206. 

61. 

$2.76. 

16. 

5413||. 

32. 

13,185|Mf- 

48. 

$3116. 

62. 

Tlie  former. 

17. 

6729^|. 

33. 

13,187Mff. 

49. 

$5.50. 

63. 

27. 

Article  88. 

2. 

10. 

3.    14. 

4.    14. 

5.    20. 

6.    10.             7.   6. 

Article  89. 

1. 

35. 

3.    1. 

5. 

29. 

7.    203. 

9.   300. 

2. 

10. 

4.    H.                 6. 

4^. 

8.    93 

• 

10.   9|. 

Article  91. 

21. 

$240. 

27. 

$2.25. 

33. 

15  yr. 

38. 

88  times. 

22. 

1140  mi. 

28. 

64  yr. 

34. 

71  da. 

39. 

28^. 

23. 

6  da. 

29. 

24hr.,  5^|hr 

.    35. 

,    40^. 

40. 

rC,$  17,000; 
Id,  $6985. 

24. 

10  hr. 

30. 

$2.10. 

36. 

,    14  hr. 

25. 

$  3240. 

31. 

$7.65. 

37, 

.    $1722. 

41. 

$156. 

26. 

284  A. 

32. 

$11.58. 

ANSWERS.  415 

Article  99. 

13.  3,  3,  7.                         20.    2,  2,  2,  2,  3,  3,  13.     27.  3,  3,  6,  7,  11. 

14.  2,  2,  3,  7.                    21.    2,  2,  2,  2,  5,  5,  7.       28.  2,  2,  2, 2, 3,  3,  7,  13. 

15.  2,  5,  5,  5.                    22.    2,  3,  5,  7,  11.              29.  2,  3,  5,  5,  7,  11. 

16.  2,  3,  5,  7.                    23.    11,  13,  17.                  30.  3,  3,  5,  5,  7,  11. 

17.  2,2,3,53.                  24.    2,3,5,7,11.              31.  2,2,2,2,2,2,2,503. 

18.  2,  2,  2,  2,  3,  3,  5.       25.    7,  7,  11,  13.                32.  3,  3,  31,  37. 

19.  2,  2,  3,  131.  26.   2,  3,  3,  5,  5,  7. 

Article  102. 

2.  4788.                  8.    7|.                     14.   200  da.  20.    4f|. 

3.  2.                         9.    12  bu.                15.    $62.50.  21.    80. 

4.  18.                     10.    20  yd.               16.    80.  22.    18. 

5.  5^.                     11.    21^.                  17.    100  bu.  23.    1^  jar. 

6.  3f                     12.    llHbu.           18.    40  1b.  24.    5.tons. 

7.  /j.  13.   25  sacks.  19.    25  cords. 

Article  107. 

2.  12.       4.   15.       6.  12.       8.   16.       10.  8.      12.   15.      14.  11.      16.  9. 

3.  21.       5.  56.       7.  20.       9.   12.       11.  2.       13.  9.  15.  68. 


Article  108. 

18.  270.  21.   24.  24.    108.  27.    140.  30.    9. 

19.  60.  22.    56.     •  25.   60.  28.    144. 

20.  21.  23.    38.  26.    60.  29.    45. 

31.  16  bushels  ;  Wheat,  2  boxes  ;  Barley,  3  boxes  ;  Oats,  8  boxes. 

Article  114. 

2.  270.         4.    240.         6.    720.  8.    420.  10.    7560.  12.    630. 

3.  36.  5.    756.  7.    72.  9.    420.  11.    1400.  13.    192. 
14.    60  quarts  ;  15  ;  12  ;  10.                  15.    40  minutes  ;  8  ;  5  ;  4. 

Article  115. 

2.  5,  5,  7,  29  ;  2,  2,  2,  3,  3,  7,  19 ;  2,  .3,  3,  5,  5,  7  ;  2,  3,  7,  11,  13. 

3.  36.  4.   2,  3,  5,  11,  7  ;  13,  7,  7,  3,  3,  3 ;  19,  5,  3,  3,  2,  2,  2. 

5.    2,  7,  11,  17,  31.  6.  7.  10.    96.  11.    44.  12.    4. 

13.    30.  14.    58.  15.   90.  16.    101.  17.   42. 

18.    2  A.  ;  7  lots  ;  9  lots  j  11  lots. 


416 


ANSWERS. 


20.  42,336.  22.  5040.  24.  9240.  26.  5040. 

21.  1260.  23.  36,086.  25.  97,020.  27.  $120. 

28.  $  1.50  ;  30  nickels,  16  dimes,  6  quarters,  50  3-cent  pieces. 

29.  1  hr.  32  miii.  35.  84f.  39.  60|f.  43.  4  da. 

30.  3168  feet.  36.  224.  40.  If.  44.  140  1b. 

33.  18.  37.  6f  41.  90  bu.  45.  30  1b. 

34.  10.  38.  1^.  42.  ^cord.  46.  8  cents. 


Article  126. 

c 

31. 

I|. 

33.    ^ 

I 

35.    ^-h- 

37. 

t¥i-          39. 

m-          41. 

Mi 

32. 

M- 

34.    t\%- 

36.    x¥o- 

38. 

m-          40. 

HI-          42. 

%m 

Article  127. 

23. 

f 

30. 

if- 

37.   tIi 

44.    i. 

50.    \\. 

24. 

f 

31. 

m 

38.    \l. 

45.    M|. 

51.    f 

25. 

i- 

32. 

h 

39.  m- 

46.    \. 

52.    I. 

26. 

A- 

33. 

H- 

40.    x^3- 

47.    f 

53.    i. 

27. 

if- 

34. 

U- 

41.    h- 

48.    iJ. 

54.    f 

28. 

i 

35. 

ii^ 

42.    f 

49.    f 

65.    M- 

29. 

i- 

36. 

f 

43.    i. 

Article  128. 

41. 

1^. 

49. 

¥/ 

56.    -W- 

63.   W- 

70.   ^'^5. 

43. 

^p. 

50. 

¥i^ 

57.    ^K 

64.    14^35.. 

71.    -^LSjOA 

44. 

W. 

51. 

^1^ 

58.    AfJ^. 

65.    ^^K 

72.    ^fl^; 

A235 

45. 

W-. 

52. 

^01 

59.    Yi^. 

66.    4||i 

73.  ^xV-;  ¥^- 

46. 

¥^. 

53. 

¥f 

60.    W. 

67.   Hl^. 

74.     $i.9^^Q, 

47. 

13^51_ 

54. 

W 

61.    Y/. 

68.    ^Vi^- 

75.    HP- 

48. 

If^. 

55. 

131 

62.   V^. 

69.   -^/j^. 

Article  129. 

33. 

20|. 

40. 

20f. 

47. 

60f. 

54.    1490||. 

34. 

m- 

41. 

29|. 

48. 

74M. 

55.    1647  fV 

35. 

iif. 

42. 

9/,. 

49. 

92||. 

56.    1326|f. 

36. 

3t\. 

43. 

19i|. 

50. 

31|. 

57.    1000. 

37. 

m 

44. 

lOff. 

51. 

159^. 

58.    250. 

38. 

2. 

45. 

21. 

52. 

18^. 

59.    292|f. 

39. 

5^. 

46. 

im- 

53. 

53^. 

60.    1393ii 

61.    123  bu. 

ANSWERS.  417 

» 

Article  131. 

2.  ^,  A.  7.  M,  H-  12.  M,  M,  H. 

3.  li  H-  8.  M,  fi.  13.  M,  M,  l|. 

4.  A,  A.  9.  if,  ^5,  t',.  14.  f^\,  ^0^,  ^. 

5.  ft,  ft.  10.  ,\,  ^%,  i§.  15.  ft,  60,  ^|. 

6.  ii  ii  11.  H,  ii  M-  16-  IS,  IS,  ti  M. 

17.  M,  H,  H,  M-  27.  M,  H,  ^TT- 

18.  ttVt^,  ^^%.  Ul  m-  28.    ^^„  ^<f3,  /^o^. 

19.  MM,   HIS,  mi  mi  29.    If,  e,  ^%,  /t7- 

20.  lie,  fsis,  mh  lui  30.  Hi  f§-g,  III,  te§. 

21.  H,  ti  n,  fs.  31.  ,^^,  tV^,  ,u,  m- 

22.  Ill,  Hi  Ml,  Iff-  32.  Hi  in,  m,  m- 

24.  M,  fi,   li  33.    li,  ^V  if,  ij^. 

25.  H»  ll>  ^-  34.    I  is  larger. 

26.  if,  li  ^^. 

Article  132. 

21.  If  30.  Iflf.  39.  28J|.  47.  $117f. 

22.  HI.  31.  2f^\.  40.  28f^.  48.  $39f 

23.  li|.  32.  lO^V^-  41.  75||.  49.  45^^  hr. 

24.  If.  33.  lOff.  42.  65f^.  50.  llff  tons; 

25.  2^.  34.  9fi  43.  78|^  mi.  $65^^. 

26.  2^^.  35.  6f|.  44.  111/^  A.  51.  IQl^f  lb. 

27.  21  36.  8.  45.  112|  yd.  52.  45|  tons. 

28.  If.  37.  11||.  46.  41^V<jrods.  53.  lOOf  cords. 

29.  2/5.  38.  5||. 


Article  133. 

26.  ^j.  35.    14f  44.    ff.  53.  4006t55. 

27.  i  36.    13i|.  45.    ^%  54.  642^^. 

28.  xV-  37.    H-  46.    iVu-  55.  lOf  gallons. 

29.  I  38.    Ji  47.    Jj.  56.  4J  rods. 

30.  15|.  39.   ^iff.  48.    y\%-  57.  25f  gallons. 

31.  3.  40.    If  49.    }|.  58.  121^  bu. 

32.  2f.  41.    II  50.   901f  59.  84yV!j. 

33.  5^.  42.  ^.  51.   4001tV  60.  17^9^  yards. 

34.  13|.  43.    l^.  52.    118^^  61.  $4|. 

62.  $5^. 


418  ANSWERS; 


Article  134. 

20.  If.  34.    15.  48.    36.  63.   257f.  78.   477^. 

21.  |.  35.    17|.  49.    9|.  64.    3894.  79.    1197. 

22.  ^^.     ^  36.    6.  50.   9|.  65.    505."  80.    959. 

23.  i.  37.    71.  51.    205.  66.    632^.  81.    25711. 

24.  1.  38.    322.  52.    309.  67.    6891.  82.    376. 

25.  j%.  39.    17 1.  53.    72.  68.    3073.  83.    504|. 

26.  |.      .  40.    2^.  54.   98|.  69.   4303|.  84.    248. 

27.  f.  41.    11.  55.    323.  70.    5247.  85.    996f. 

28.  35.  42.    9|.  56.    750.  71.    1350.  86.    13,676. 

29.  l  43.    396.  57.    1818.  72.    2151.  87.    62,054. 

30.  f.  44.    8|.  58.    1428.  73.    11||.  88.    48,510. 

31.  ^V  45.    12f.  60.    55^.  74.    85|.  89.   47,250. 

32.  ^V  46.    61 1.  61.    188.  76.    498^.  90.    150,994f. 

33.  If.  47.    32.              62.    84f            77.    171.  91.   f ;  2il. 
92.    H;if  93.    31;  2.         94.    lj%.        95.    $35§i.  96.    $115^^0. 
97.   $30i|.  98.    Elder,  74|^  A.,  Younger,  492  9  A.  99.    1515  days. 
100.    j%.       101.    540  boys.        102.    $709j?5.       103.    $3620.       104.    ^j%. 
105.    $20 5V                106.    $102.                 107.    $23.31.  108.    336. 


Article 

135 

• 

18. 

H. 

27. 

m- 

36.    f. 

46. 

83ff. 

57.    79|. 

19. 

ItV 

28. 

2i 

37.  m- 

48. 

6|i 

58.    88ff-. 

20. 

M. 

29. 

llf. 

39.    9^-,. 

49. 

76f|. 

59.    75^f. 

21. 

f. 

30. 

12. 

40.    j%. 

50. 

35i§. 

60.    28^. 

22. 

H- 

31. 

jh- 

41.    T^. 

51. 

77^V 

61.    12,451|. 

23. 

3|. 

32. 

■h- 

42.   2. 

52. 

1554^J, 

62.    8014. 

24. 

llf. 

33. 

h 

43.    21. 

53. 

5981f. 

63.   8279^V 

25. 

m- 

34. 

4f. 

44.    jh. 

54. 

2909i|^.     64.    3772ffi. 

26. 

h 

35. 

3. 

45.      33^V2 

. 

55. 

3879^. 

,       65.    $,V.. 

66. 

$7' 

172H. 

67.    $111. 

68. 

65^^. 

69. 

574^1  yards. 

76. 

10  hours. 

84. 

Hi- 

91.    12|rods. 

70. 

10  days. 

77. 

^M 

\. 

85. 

H- 

92.    14  tons. 

71. 

•$4f. 

78. 

10  books. 

86. 

5. 

93.    $2. 

72. 

m- 

80. 

12. 

87. 

32. 

94.    37  bags. 

73. 

44  barrels. 

81. 

23f 

88. 

1^- 

95.    100  bushels. 

74. 

80  tablets 

5. 

82. 

SB. 

89. 

4^\. 

96.   f. 

75. 

160  pounds. 

83. 

^\. 

90. 

$120^. 

97.    $75. 

ANSWERS.  419 


Article  136. 

6.  9.                      17.    40.                    28.    28.  39.    $15. 

7.  10.                     18.    f                      29.    209.  40.    f. 

8.  16.                    19.    |.                      30.    f  41.    $8400. 

9.  9.                      20.    |.                      31.    20.  42.    $4890|. 

10.  12.                    21.    f.                      32.    42.  43.    $490. 

11.  25.                    22.    I.                      33.    56.  44.    $5000. 

12.  20.                    23.    f.                      34.    f.  45.    f. 

13.  6.                      24.   49.                    35.    70.  46.    135  sheep. 

14.  15.                    25.    f                      36.   f  ;  f  47.    756  yards. 

15.  21.                    26.    .39.                    37.    280  sheep.  48.    f. 

16.  30.                   27.   ^^.               '    38.    $16.  49.    ^. 

50.    $7200. 

Article  137. 

31.  300  bu.                        49.    32^.                           67.  ^%,    ||,    ^|^,    4^, 

32.  19.                               50.   $100,375.  ^,  Mh  iH- 

33.  if;  ^i^  greater.         51.    $524,875.                   68.  If  f  da. 

34.  18  da.                          52.    3fg  A.                         69.  2f|  hr. 

35.  $60.                             53.    $8.5.375.                     71.  $52.  .50. 

36.  $18.                             54.    $.95.                           72.  2j%\. 

37.  $62.                             55.    Increased  3*j.             73.  29  1b. 

38.  $89.70.                        56.    54^^^  A.                     74.  7. 

39.  $.09.                           57.    I                               75.  $1,125. 

40.  $1.1H.                        58.    9|.                               76.  Son,      $24,000; 

41.  3|f.                              59.    1428.  Daughter,  $15,000. 

42.  9||.                              60.    tIi-                             77.  $46,293.33^. 

43.  93%  tons.                     61.    2f|.                             78.  $800. 

44.  2^1.                             62.    17i-3,:V                         79.  $^^. 

45.  9433^  A.                       63.    j^V                             80.  12^Vll>. 

46.  43  mi.                          64.    7f.                              81.  21^3,. 

47.  Geo.  $^3^  greater.      65.    /^.                              82.  If. 

48.  41  cords.  66.    j\%. 

Article  144.     Decimal  Fractions. 

1.  Seven  tenths.  3.    Seven  thousandths. 

2.  Seven  hundredths.  4.    Seven  hundred-thousandths. 

5.  Three  thousand  sixty -five  hundred-thousandths. 

6.  Sixteen  thousand  nine  hundred  eighty-four  hundred-thousandths. 


420  ANSWERS. 

7.  Ten  thousand  sixteen  hundred-thousandths. 

8.  Fifty-four  ten-millionths. 

9.  Thirty-five  and  eighteen  thousand  six  hundred-thousandths. 

10.    Five  ten-thousandths.  11.    Five  hundred-thousandtlis. 

12.  Four  and  ninety-eight  thousand  six  hundred  twenty-five  hundred- 
thousandths. 

13.  Thirty-eight  thousand  six  hundred  ninety-four  and  six  hundredths. 

14.  Nine  and  ninety-eight  million  four  hundred  sixty-three  thousand 
four  hundred-millionths. 

15.  Two  hundred  thirty-five  and  eight  hundred  fifty  thousand  sixty- 
two  millionths. 

16.  One  hundred  and  one  hundred  four  millionths. 

17.  Nine  and  one  million  six  hundred  thirty-two  thousand  two  ten- 
millionths. 

18.  Three  thousand   five  hundred  forty-three   and  four  million  five 
hundred  thirty-six  thousand  nine  hundred  eighty-two  ten-millionths. 

19.  Thirty  and  three  million  three  hundred  three  thousand  three  hun- 
dred three  ten-millionths. 

20.  Three  hundred  three   and  three  hundred  three  thousand  three 
hundred  three  millionths. 

21.  Nine  and  nine  hundred  ninety-nine  thousand  nine  hundred  ninety- 
nine  millionths. 

Article  145. 

22.  .4,  .17,  .05,  .325,  .005,  .015,  19.724. 

23.  .7504,  16.0125,  .0006,  .5000. 

24.  .17211,  .00004,  .00015,  18.00216,  .00112. 

25.  .29,  .029,  .0029,  .00029,  1.1,  1.01,  1.001,  1.0001,  1.00001. 

26.  324.000126,  4582.36242,  .000017,  .00005,  24.0003406. 

27.  .000010,  .00824, .31,  .00216, .00007846,  4.00015. 

28.  .8.  32.    .00289.  36.    .3.  40.    .000001. 

29.  .16.  33.    .028654.  37.    .01.  41.   1.000. 

30.  .615.  34.    .0000563.  38.    500.5.  42.    .00005. 

31.  .2123.  35.    15.005.  39.    .0027.  43.    .0275. 


Article  148.     Reduction  of  Decimals. 

44.  .500000,  .017000,  .125600,  .000155,  29.803000. 

45.  .80062,  305.24000,  70.50000,  3.85263. 

46.  .1000000,  .0001000,  1000.0010000,  1.0100385. 

47.  .26000,  .13682,  9.40000,  25.00000,  8.63521. 


ANSWERS. 

421 

Article  149. 

48. 

^• 

51.    1^. 

54.    ^,.           57 

.  tV.- 

"0.          34^QQ*I)Q|^. 

49. 

^' 

52.    I 

55.    UU'          58 

-  w^h- 

61.    2^TTr^\^jj. 

50. 

H- 

53.    ^. 

56.    2J.               59 
Article  150. 

'.    2S^'^. 

62.    1084^^^. 

64. 

h 

66.   |. 

68.    i 

70.   f 

72.    m- 

65. 

^' 

67.   i. 

69.    f. 
Article  152. 

71.    |. 

73. 

.8. 

77.    .1875.         81.    .83f 

85.    .425. 

89.   66.66f. 

74. 

.625. 

78.    .5|. 

82.    .2916|. 

86.    .75. 

90.    25.125. 

75. 

.75. 

79.    .6. 

83.    .875. 

87.    12.5. 

91.    16.25. 

76. 

.66f 

80.    .5. 

84.    M^\. 
Article  153. 

88.    33.33^ 

;.     92.    16.25125. 

93. 

1657.822. 

97.    161.1095. 

101. 

40314.039415. 

94. 

1914.69356. 

98.    105682.1451. 

102. 

5.55655957. 

95. 

204.474. 

99.    221.212. 

103. 

58.1933. 

96. 

1944.425. 

100.    278.1223. 
Article  154. 

104. 

2.854. 

109. 

6.14994.            114. 

.000099. 

119.    75.621. 

105. 

37.644. 

110. 

847.638.             115. 

9.9001. 

120.    999.995. 

106. 

25.05017. 

111. 

.09999.               116. 

9.825. 

121.    1.19983. 

107. 

15.2599. 

112. 

999.99.              117. 

43.698519. 

122.    7.66. 

108. 

32.15596. 

113. 

19.99795.           118. 

2.977. 

123.    .620005. 

124. 

3.21375. 

128. 

.099999. 

125. 

.3456. 

129. 

Neither. 

126. 

9.9,  9.99, 

5.02,  8.95.                          130. 

Reduce  | 

to  thousandths. 

127. 

999999.900001. 

Article  155. 

1. 

.608. 

6.    90.978.             11.    6.76. 

16.    .110889. 

2. 

.00075. 

7.    36.704.             12.    1440.45. 

17.    .0000080184. 

3. 

25.50. 

8.    1.01101.            13.    1171.9052. 

18.    957.32. 

4. 

15.2. 

9.    17.329.             14.    .1. 

19.    9.23. 

5. 

.2756. 

10.    112.073.           15.   4.0625. 

20.    1111. 

422 


ANSWERS. 


32.  17,288. 

33.  11,682. 


Article  157. 

34.  1768.4. 

35.  3,752,500. 


36.  3784. 

37.  31,240. 


38.    1. 


Article  l58. 
39.    .000049.  40.    .000099999996.     41.    181.3259. 


Article  159. 

1. 

.45. 

6. 

12. 

11.    2389.636+. 

16. 

100. 

2. 

.0006. 

7. 

331.487+ 

12.    .0066. 

17. 

.01. 

3. 

1200. 

8. 

800,000. 

13.    29. 

18. 

1,000,000. 

4. 

1. 

9. 

.25. 

14.    6000. 

19. 

.000001. 

5. 

.004. 

10. 

.763. 

15.    1000. 

20. 

.1865. 

18.  $15.40. 

19.  $7,535. 

20.  $145.86 


21. 
22. 
23. 


Article  162. 

$318.3345.         24.   $236.28. 


$20,405. 
$27.30. 


25.  $193.25. 

26.  $53.01. 


27. 
28. 
29. 


$47,364. 

$857,255. 
$  35.30409. 


Article  166. 

10.  $125. 

11.  $5^. 

Article  168. 
22.    166 1  bu.  23.   200  lb.  24.    228  doz.  25.   80  qt. 


8.  $8.25,  $24.66|,  $12.00,  $12.00,  $4.00,  $10.00. 

9.  $98.75. 


Article  169. 

8.  .1;  .24;  .379,  .1000;  .00085;  .020079. 

9.  1006.000502. 

10.  315001.0011;  38.007;  8270942.005;  1.7. 

11.  421.0005;  1027.27;  99.0000099. 

12.  .1  ;    .02  ;    .003  ;    .0004  ;    .00005  ;    .000006  ;    12.17  ;    42.32  ;    78.589  ; 
200.2001. 

13.  .At>.  h^uh,  :^,  i^^hn^  ^iku^  2%,  h  h  811,  91^5,  4001^1,  W4V 

14.  .625,    .4,    .16,     .9375,     .0468+,     .375,      20.5957+,     8.004,     4.0625, 
708.655. 


ANSWERS. 


423 


15.  .15,   .0775,    .188, 
.6465+,  .0000525,  .78875, 

16.  665.456711. 

17.  3628.52791. 

18.  61870.29177. 

19.  12387.56776. 

20.  688.6634. 

31.    594. 


.107875,    .125,    .08^,  .224,    .04504,    .3775,    .38^, 
.3811+. 

21.  910.88127.  *   26.   365.48. 

22.  406.368661.  27.    199.98. 

23.  2.51.  28.    I  is  .4|  greater. 

24.  .533.  29.    $2.16. 

25.  26.  30.    1,999,999.999998. 
0849  acres  in  all :  293.8349  left. 


32.    .9999. 

42.    62.34f  cents.              52. 

146.46786+. 

33.    2.3836. 

43.    $9.28^.                       53. 

3,360,000. 

34.    147. 

44.    $.21|.                         54. 

1.51015. 

35.    .001. 

45.    5.115365472.              55. 

32,320.03. 

36.  62.5. 

46.   417,000.                      56. 

.70458+. 

37.    181.87548. 

47.   47.6.                            57. 

8.41424+. 

38.    27.2544. 

48.    1719.523+.                  58. 

16.16366+. 

39.    109.090908. 

49.    14.                              59. 

4.5733+. 

40.    $44.74|. 

50.    122,733.333+.             60. 

.00006. 

41.    $76.80. 

51.    156.414+. 
61.    35,002  -  15  =  2333.46f. 

62.   206.52. 

63.    $3600.             64.   $120. 
Article  175. 

65.   $2.00. 

1.    $88.90. 

2.    $8,155.                3.    $410,185. 

4.    $6.39. 

5.   $143. 

6.    $102.15.               8.   $417,175. 
Article  176. 

1.    70.0542  yd. 

11. 

$15.97.           22.   204.986363. 

33.   $23.75. 

2.   29.875  yd. 

12. 

$6.76.              23.    5.0964+. 

34.    30. 

3.    27  yd., 

13. 

$317.61.          24.    10. 

35.    3000    thou- 

$.3,375. 

14. 

$66.54|.          25.    .9724. 

sandths. 

4.    2.105  yd. 

15. 

$45.70.            26.    38.996. 

36.    100. 

5.   $13,255. 

16. 

$17.68.            27.  487.541+ bu. 

37.    .0025. 

6.    $7.99. 

17. 

If                   29.    36. 

38.   $136f|. 

7.    $.98  gain. 

18. 

If.                   30.   A  and  Beach 

39.    2|§f^i. 

8.   $818.80. 

19. 

y«^\.                        $200,  C  $386. 

40.    i^-,. 

9.   $56.80. 

20. 

\\.                   31.    18. 

41.    $3972.90f 

10.    $13.11. 

21. 

$64,000.          32.    .36. 

42.    $16,000. 

43. 

16  children.                          45.    Increased  ^§5. 

44. 

A  $10,  B  $15.                     46.    diminished  -^. 

47.   2856. 

18.   25.                        50.   m. 

51.  24. 

424 


ANSWERS. 


2.  646  in. 

3.  773  ft. 

4.  204,978  in, 

5.  81,701  sq.  yd. 

6.  25,679,196  sq.  in. 

7.  5045  links. 

8.  790,596  cu.  in. 
^  9.  1036  cu.  ft. 

10.  127  pt. 

11.  33,805  minims. 
^12.  507  pt. 

13.  528  qt. 

14.  146,794  grains. 

15.  73,774  oz. 

16.  94,851  grains. 

17.  101,658  sec. 

18.  18,035  far. 


Article  209. 

19.  2,400,540  sec. 

20.  1,817,332  sec. 
40  quires. 
672  pt. 
12,816. 
87,648  hr. 
891  in. 

26.  $197.12. 

27.  1296  sq.  in.; 
4356  sq.  ft. ; 
46,656  cu.  in. 
696  hr. 
1155  cu.  in. 
12,390,400  sq. 
108,900  sq.  ft. 
36  oz. ;  48  oz. 
$2.40  loss. 


21. 
22. 
23. 
24. 
25. 


28. 
29. 
30. 
31. 
32. 
33. 


34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 
yd.  47. 
48. 
49. 
50. 


35,840  oz. 

$9. 

18  centuries. 
2  da.  2  lir. 
180  degrees. 
2160  degrees. 
$24. 

225,932  in. 
35,640  ft. 
19,138,464. 
76,2051  cu.  in. 
7424  cu.  ft. 
69,056  oz. 
21,972  gr. 
1947  gi. 
24,000  sheets. 
39,180  min. 


3. 

5. 

7. 

9. 
10. 
11. 
12. 
13. 
14. 
20. 
22. 
24. 

1. 
2. 
3. 
4. 
5. 
6. 
7. 


Article  210. 

3  mi.  4  fur.  20  rd.  5  yd.  2  ft.  8  in.        4.    6  mi.  240  rd. 

3  A.  28  sq.  rd.  5  sq.  yd.  3  sq.  ft.         6.    16  cu.  yd.  9  cu.  ft.  3  cu.  in. 

58  cd.        8.  2  T.  3  cwt.  16  lb. 

3  lb.  7  oz.  7  dr.  16  gr.  Apoth. 


60  gal.  3  qt.  3  gi. 

5  bales. 

3  wk.  6  da.  5  hr. 

£6  2  far. 

23°  30'  23". 

17  T.  16  cwt.  82  lb. 

21  bu.  2  pk.  4  qt. 

28||  cords. 

$  13.20. 
5220  minims. 
$2.80. 

120,300  times. 
50.5  degrees. 
30  degrees. 

(a)  $1.17i; 

(b)  $515f. 


3  lb.  7  oz.  18  pwt.  4  gr.  Troy. 

15.  8  pt.,  1  flu.  oz.  7  flu.  dr.  1  minim. 

16.  1  mi.  50  ch.  25  links. 

17.  560  bu.  1  qt. 

18.  5  lb.  7  oz.  16  pwt.  19  gr. 

19.  6  lb.  10  oz.  4  dr.  1  sc.  12  gr. 
21.  43  Gr.  gro.  10  gr.  7  doz.  4  pens. 
23.  3  sq.  rd.  17  sq.  yd.  1  sq.  ft.  55  sq. 


8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 


Article  211. 

.    $297.12. 
■    2iffmi. 

27^  ft. 

40,224^  ft. 

3/2¥tr  mi. 
48  qt. 
6pt. 
$2.60. 


16. 
17. 
18. 
19. 
20. 
21. 
22. 
4^  mi. 


319  pt. 

93  boxes. 

$3.60,  gain. 

$9.60. 

67  cents. 

$1.41,  gain. 

30  mi.  ;    2J  mi. ; 


ANSWERS. 


425 


24. 
25. 
26. 

27. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
.488 
12. 
13. 
14. 


4. 
5. 
6. 
7. 
8. 
9. 
10. 


^1.28. 

58|  cents. 
$1.20. 
10  cents. 
$  12.96. 


29. 
30. 
31. 


21  forks. 
6pt. 
1280  rd. 
$1.05. 


32. 
33. 
34. 
35. 


160  rd. 

56  ft.  or  18|  yd. 

9^  ft. 

228  ft. 


213  rd.  1  yd.  2  ft.  6  in. 
133  sq.  rd.  10  sq.  yd.  108  sq.  in. 
8  oz.  11  pwt.  10^  gr. 
8  cwt.  57  lb.  2  oz.  4f  dr. 
2  qt.  3^  gi. 

I  fur.  31  rd.  1  ft.  10  in. 

II  da.  6  hr. 
1  qt.  2  gi. 

68  sq.  rd.  8  sq.  yd.  2  sq.  ft. 
sq.  in. 

4  cwt.  72  lb.  12  oz.  12.8  dr. 
10  degrees  53  min.  24  sec. 
4  fur.  5  rd.  1  yd.  3.6  in. 


Article  213 
15. 
16. 
17. 


3  fur,  33  rd.  3  yd.  lOf  in. 
5  mo.  4^  da. 
2  qt.  I  pt. 

18.  3  oz.  4|ff  dr. 

19.  102  sq.  rd.  25  sq.  yd.  8  sq.  ft. 
51f  sq.  in. 

20.  2  pk.  5  qt.  l|f  pt. 

21.  43  gal.  1  pt.  2^  gi. 

22.  86  sq.  rd.  4  sq.  yd.  5  sq.  ft. 
127^5^  sq.  in. 

23.  6s.  8d. 

24.  1  fur.  13  rd.  1  yd.  2  ft.  6  in. 


f  mi. 

f  A. 

f  yr. 

fib. 
f  T. 

.43748  mile. 
.0533+  month. 
.3125  gal. 


11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 


Article  214. 
.4267  A. 
.61625  lb. 
.361  year. 

I  mile. 
l%l  month. 
.24  T. 
.3824  cord. 


19. 
20. 
21. 
22. 
23. 
24. 
25. 


.03026  circle. 
.61626  mile. 
Alb. 
If  gal. 
jh  mile- 
r^  cord. 
tV?  gr.  gro. 


To  find  what  part  one  denominate  number  is  of  another  : 

2.  t^^.  5.    .7586+.  8.    ^\V  10. 

3.  flif  6.    .6420+.  9.    £^.  11. 

4.  AVW-  7.    B|. 


Til- 

.4964+. 


Article  215. 


4.  102  T.  1  cwt.  84  lb.  12  oz. 

5.  77  deg.  16  min.  38  sec. 

6.  806  sq.  yd.  3  sq.  ft.  137  sq.  in. 

7.  436  yr.  290  da.  20  hr.  44  min. 
16  sec. 


8.  390  bu.  1  pk.  2  qt.  0  pt. 

9.  19  cords  3  cd.  ft.  13  cu.  ft 

10.  18  T.  2  cwt.  92  lb.  16  oz. 

11.  50  hr.  39  min.  34  sec. 

12.  31  mi.  197  rd.  6  yd.  6  in. 


13. 

23  A.  132  sq.  rd.  30  sq.  yd.  2 

titSXO* 

sq.  ft.  56  sq.  in. 

14. 

62  cords  5  cd.  ft.  3  cu.  ft. 

19. 

67°  33'  46". 

15. 

19  T.  11  cwt.  15  lb.  6  oz. 

20. 

11  mi.  43  rd.  5  yd.  2  ft. 

16. 

501  mi.  28  rd.  2  yd.  1  ft.  6  in. 

21. 

167  rd.  2  yd.  0  ft.  2^  in. 

17. 

64  gal.  1  qt. 

22. 

12  bu.  3  pk.  4  qt.  j\  pt. 

18. 

46°  2'  20''. 

23. 

12  cwt.  56  lb.  2  oz.  14|  dr. 

Article  216. 

2. 

5  A.  104  sq.  rd.  2  sq.  ft. 

12. 

1  bu.  2  qt. 

3. 

4  hr.  36  min.  40  sec. 

13. 

8  oz.  2  dr.  9  gr. 

4. 

22  gal.  3  qt.  2  gi. 

14. 

48°  25'  37". 

5. 

2  A.  3  R.  12  sq.  rd.  24  sq.  yd. 

15. 

8  mi.  163  rd.  4  yd.  1  ft.  6  in. 

Isq.: 

ft.  36  sq.  in. 

16. 

197  rd.  1  ft.  4|  in. 

6. 

64  da.  21  hr.  29  min.  48  sec. 

17. 

10°  8'  26". 

7. 

14  T.  19  cwt.  49  lb.  14  oz. 

18. 

9|oz. 

8. 

236  mi.  13  rd.  5  yd.  2  in. 

19. 

14  bu.  3  pk.  1  qt.  f  pt. 

9. 

4  sq.  rd.  8  sq.  yd.  8  sq.  ft.  36 

20. 

4  lb.  12  oz. 

sq.  in 

21. 

1  yd.  1.4  in. 

10. 

2  cwt.  85  lb.  4  oz. 

22. 

6  da.  22  hr.  20  min. 

11. 

26  gal.  3  qt.  1  pt. 

Article  217. 

3. 

155  yr.  6  mo.  23  da. 

9. 

$  142.50. 

4. 

3  yr.  11  mo.  25  da. 

14. 

116  da. 

5. 

67  yr.  9  mo.  22  da. 

15. 

May  5,  1907. 

8. 

138  da. 

16. 

595  min. 

Article  218. 

2.  160  gal.  1  qt.  1  pt.  3  gi.  102  10.  754  mi.  120  rd. 

bu.  2  qt.  4  pt.  11.  65  oz.  15  pwt.  15  gr. 

3.  607  mi.  169  rd.  11|  ft.        12.  363  bu. 

4.  451  A.  138  sq.  rd:  29  sq.  yd.    13.  400  mi.  244  rd.  14  ft. 

5.  55  T.  10  cwt.  14.  21  hr.  20  min.  44  sec. 

6.  44  lb.  9  oz.  15.  53  gal.  2  qt.  1  pt. 

7.  248  gal.  2  qt.  16.  45  A.  10  sq.  rd.  17  sq.  yd.  4 

8.  $228.75.  sq.  ft.  72  sq.  in. 

9.  18  lb.  12  oz. 

Article  219. 
8.  24  bu.  1  pk.  If  pt.  7.  75  A.  49  sq.  rd.  25  sq.  yd. 

4.  60  mi.  240  rd.  16  ft.  8.  1  bu.  2  pk.  7  qt. 

6.  3  lb.  11  j%  oz.  9.  10  A.  14  sq.  rd.  5  sq.  yd.  6  sq.  ft, 

a  25  doz.  10.  3  mi.  Ill  rd.  2  yd.  1  ft. 


ANSWERS. 


427 


11.  16  min.  47  sec. 

12.  3  qt.  1  pt.  3  gi. 

13.  11  mi.  35  rd.  3  yd.  2  in. 

14.  27|da. 

15.  15  packages. 

16.  66|  sq.  rd. 

17.  12  T.  16cwt.  901b.  15  oz. 

18.  Tifbu. 

19.  35^^  packages. 

20.  54^  da. 

21.  58ff  sacks. 

22.  171  ill  jars. 

23.  13  hr.  57  miu.  37f  sec. 

24.  26  oz.  6  dr.  2  sc.  7  gr. 


25.  168  cords,  5J  cd.  ft. 

26.  917  T.  4  cwt.  70  lb.  12  oz. 

27.  6  bu.  1  pt. 

28.  80  T.  6  cwt.  50  lb. 

29.  2  gal.  3  qt. 

30.  12  A.   14  sq.  rd.   12  sq.  yd. 
2dr.l29|  sq.  in. 

31.  81  mi.  22  rd.  4  yd.  1  ft.  3|  in. 
148  gal.  3  qt.  1  gi. 
3  lb.  4  oz.  8  pwt.  9f  gr. 
745  mi.  126  rd.  |  ft.  or  4f  in. 
17  bu.  2  qt.  If  pt. 
1  yr.  6  mo.  17  hr.  14  min.  25 


32. 
33. 
34. 
35. 
36. 
sec. 


1.  2  and  7. 

2.  3  X  8  ;  6  X  4  ; 
2x2x2x3. 

3.  3,  3,  5,  7,  7. 

5.  7  yd. 

6.  2520. 

7.  15. 

8.  180. 

9.  3  yd. 

10.  $4.50. 

11.  \. 

12.  4  times. 

13.  $300. 

14.  $192. 

15.  24  ft. 

16.  1280||. 

31.  113.0976o 

32.  176,715.         4 

33.  314.16. 

34.  706.86. 

35.  452.3904. 

36.  804.2496. 

37.  198,943+. 


Article  220. 

17.  680f  A. 

18.  AV^. 

19.  9,699,690. 

20.  50  A. 

21.  36  bu.  3  pk.  6  qt. 

22.  89  lb.  2  oz.  6  pwt. 
4gr. 

23.  m- 

24.  f 

25.  277  bales,  2  bun- 
dles, 10  quires,  18  sheets. 

26.  241  rd. 

27.  480  A. 

28.  2/^  mi. 

29.  6493/x. 

30.  5600  rails. 


31. 

32. 

33. 

34. 

35. 

36. 
19  gr. 

37. 

38. 
■    39. 

40. 

41. 

42. 

43. 

44. 
7igr. 


14701  ft. 

3138.135  Stat.  mi. 

33  mi. 

57.6  sq.  rd. 

6  lb.  1  oz.  17  pwt. 

.39^  or  .3916+. 

3  lb.  10.84  oz. 
$  254.826. 
.7955+. 

13  bu.  3  pk.  4  qt. 
16  lb.  7  oz.  13  pwt. 


Article  233. 

38.    20,106.24. 
12076.3104. 
7854. 
40  sq.  yd. 
180  area. 
78.54  sq.  ft. 


39. 
40. 
41. 
42. 
43. 


44.   157.08  ft.  diam. 


45.  37.6992    ft.    c 
cumference. 

46.  12^  ft.  radius. 

47.  87^  A. 

48.  28^!^  A. 

49.  140  rd. 

50.  7272  sods. 


4ii8 

ANSWERS. 

61.    181^  ft.  deep. 

57.    18.4+sq.  rd. 

62. 

36  rd. 

52.    1620  tiles. 

58.    3.183+ ft. 

63. 

300  sq.  ft 

53.    14,400  shingles. 

59.    9.4248  in. 

64. 

84  sq.  ft. 

54.   49  sq.  yd. 

60.    50.92+ A. 

65. 

24  sq.  in. 

55.    688  sq.  yd. 

61.   Semicircular  plat 

56.   21  min. 

has  twice  the  area. 
Article  234. 

3.    $22.20. 

9.    61  yd. 

17. 

$43.80. 

4.    45  sq.  yd. 

10.    $36. 

.     18. 

$1.80. 

5.    5  strips ;  6  strips ; 

11.    45  yd. 

19. 

^^  yd. ; 

6  strips. 

12.    28  yd. 

turned  under. 

6.    6  strips ;  7  strips ; 

13.    64  yd. 

20. 

$91,875. 

8  strips. 

14.    34  yd. 

21. 

430  yd. 

7.   26|  yd.  in  length. 

15.   42  yd. 

22. 

21    yd.    be 

8.    38  yd. 

16.    45^  yd. 

23|  yd.  carpet. 

Article  235. 

2.    $87.42i 

7.    $27.40. 

11. 

$24.16f. 

3.    174|sq.  yd. 

8.    $58.50. 

12. 

$  167.84f 

4.    $4.00. 

9.    $30.32. 

13. 

$7,565. 

5.    $26.40. 

10.    $26.60. 

14. 

87f  sq.  yd. 

6.    SS^sq.yd. 

Article  236. 

1.    $6.75. 

3.    12  rolls. 

5. 

$8.10. 

2.   36  strips ;  6  rolls. 

4.    15  rolls. 
Article  238. 

6. 

11  rolls. 

1.   18|bd.  ft. 

5.    $25,844. 

10. 

240  bd.  ft. 

2.    18f  bd.  ft.  ;   37^ 

6.   420. 

11. 

$7.84. 

bd.  ft. ;  28J  bd.  ft. 

7.   $4.80. 

12. 

$6,804. 

3.    1584  bd.  ft. 

8.   $3.84. 

13. 

$1,425. 

4.    704  bd.  ft. 

9.    $23.04. 

14. 

$13,944. 

Article  239. 

» 

1.    $6.76. 

6.    $146.64^. 

9. 

21|  rd. 

2.    $22.50. 

6.    $48. 

10. 

36  yd. ;  32 

3.    $7.42^. 

7.    $11.53. 

11. 

$43.89. 

4.   $109.97. 

8.    $33.33|. 

12. 

$82.40. 

i  yd. 


border ; 


ANSWERS. 

42S 

13. 

8Jch. 

17.    7isq. 

ft.                     21. 

131i  sq.  ft. 

14. 

56.5488  ft. 

18.    7957  i§i 

mi.                  22. 

25,142f  mi. 

15. 

28.2744  sq.  yd. 

19.    2Hft. 

23. 

450  ft.  ;   $  18.00. 

16. 

10084.03+  revo- 

20.  64  sq. 

yd.                   24. 

$12. 

lutions. 

Article  245. 

2. 

12,288  cu.  ft. 

8.   2|ft. 

14. 

.7395  cu.  in. 

3. 

Oft. 

9.    64,800  bricks.             15. 

5A¥^cu.ft. 

4. 

4950  lb. 

10.  600  blocks.                 16. 

24  sq.  ft. 

5. 

12  in. 

11.    137^V..                      17. 

2714.3424  sq.  in. 

6. 

90  cu.  ft. 

12.    30||  sq.ft.                  18. 

2111.1552  in. 

7. 

$21. 

13.   24,147^  cubes.            19. 

$38.40. 

Article 

246. 

1. 

4|  cords. 

4. 

3^  cords. 

7.   40^  cords. 

9.    $9.23. 

2. 

18f  cords. 

5. 

$10,625. 

8.    2^-i-g  cords. 

10.    64;  96;  48. 

3. 

1^^-s  cord. 

6. 

6|ft. 

Article  247. 

2. 

2393.76+. 

5. 

15.06  bu. 

8.    lOOOff 

11.    27.97fbbl. 

3. 

96.42  bu. 

6. 

3.73+  ft. 

9.    1357f. 

12.    3.11+ ft. 

4. 

51.56+  bu. 

7. 

897H-. 

10.    59|fbbl. 

13.   6.73+  ft. 

Article  250. 

11. 

1.7°  54'  55". 

21.    72°  2' west. 

12. 

76°  20'  15". 

22.    Set  back  1  hr.  16  min.  47  sec. 

13. 

3  min.  59  sec 

.  past  7*A.M. 

23.    East  22^  degrees.     ' 

14. 

847.664  mi. 

24.    57  min.  58  sec.  past  7  a.m. 

15. 

45°  west. 

25.    30  days. 

16. 

Ihr. 

26.    93°  48'. 

17. 

3  hr.  fast. 

27.    120°. 

18. 

1  hr.  11  min. 

39|i 

sec. 

28.    13°  23' 42": 

East. 

19. 

32  min.  1  sec 

.  past  11  A.M. 

29.    122°  26'  15"  West. 

20. 

36  min.  25^  sec.  past  9  a.m. 

Article 

264. 

6. 

6.3071  Km. ; 

630,710  cm. 

12.    .0086  Mm.; 

86,000  mm. 

7. 

1220.47  in. 

13.    75006.2  m. 

;    7,500,620  cm. ; 

8. 

27,685,329  mm. 

7.50062. 

9. 

32808.33^  ft. 

14.   3  Mm.   7  Hm.   6  Dm.  9  m. 

10. 

70000.006  meters. 

5  dm.  4  cm. 

3  mm. 

11. 

7500  cm. 

16.   621.3+  mi. 

30 

ANSWERS. 
Article  271. 

1. 

Sq.  Dm.            3. 

Sq.  Hm.            5.   Two. 

7.   Four. 

2. 

Sq.  m.        •       4. 

Two  places.      6.   Two. 

8.   .6556  Ha. 

9. 

33,330,000  ca. 

15.   93,800,000 

sq.  cm. 

10. 

9380  sq.  m. 

16.    l(50,7i 

50  sq 

,  cm. 

11. 

93.80  A. 

17.    16,07i 

5,000 

sq.  mm. 

12. 

.9380  Ha. 

18.    .0000160750  sq.  Km 

13. 

93.80  sq.  Dm. 

19.   2^  A. 

14. 

.9380  sq.  Hm. 

Article  275. 

3. 

7,000,000,000. 

6.    16.2  cu.  m. 

9. 

1,800,000  cu.  cm. 

4. 

.000000005. 

7.    16.  2  steres. 

10. 

1800  cu.  dm. 

6. 

One. 

8.    16,200,000,000  mm. 
Article  279. 

11. 

37.5  steres. 

5. 

.0015467  Kl. 

8.    27,200  J   272  HI.  ; 

9. 

5|  dm. 

6. 

1234.6  dl. 

27.2  Kl. 

10. 

1000  1. 

7. 

1000  1. 

Article  281. 

9. 

74,200,000  eg. 

12.    2.2+ lb. 

14. 

1,000,000  gr. 

10. 

5,430,000,000  Mg 

:.       13.   220+ lb. 

16. 

14,400  Kg. 

11. 

16,432  gr. 

Article  283. 

2. 

^f^. 

29.   Ten-millionths. 

48. 

Ih 

4. 

mh 

30.   $18.10. 

49. 

$4.63+. 

6. 

$  14,400. 

34.    2  yr.  7  mo.  29  da. 

50. 

$  144.64. 

7. 

92f|. 

35.    133^^. 

51. 

2692||f. 

8. 

/^V 

36.    16,6751. 

62. 

845f. 

9. 

mim;^' 

37.    ISs.id.;    5  cwt. 

53. 

8cd. 

10. 

$48f. 

5  lb.  1.92  oz. 

54. 

76  T.  1  cwt.  7  lb. 

11. 

51  f  days. 

38.    .39375. 

55. 

1,568,160; 

12. 

^h- 

39.   9  bu.  3  qt.  ^  pt. 

3,136,320. 

13. 

Increased  ^j. 

40.    1561  gal.  3  qt.  1  pt. 

56. 

40  cd. 

14. 

mn  hr. 

41.    1383.2  miles. 

57. 

$10.85. 

16. 

$71.68. 

42.    9  cwt.  33  lb.  5i  oz. 

68. 

7  fur.  13  rd.  14  ft. 

17. 

23.114931. 

43.    21da.21hr.36min, 

6  in. 

19. 

$63.17. 

44.    2  oz.  13  pwt.  8  gr. 

59. 

1  mi.  286  rd.  5  yd. 

20. 

265.006110. 

45.   fffofamile. 

6  in. 

21. 

1- 

46.  imm- 

60. 

14^  bu. 

26. 

.00529+, 

47.   ^. 

61. 

U' 

ANSWERS. 

431 

62. 

$40.92f. 

76. 

54f  yd. 

89. 

$14.82. 

63. 

$14,925. 

77. 

$31.16|. 

90. 

5    min.    29f    sec 

64. 

3  c.  2°  6'  4". 

78. 

80  bd.  ft. 

past 

11    A.M.  ;    54    min 

65. 

972,000. 

79. 

$5.76. 

30^  sec.  past  12  m. 

66. 

900°;  75°. 

80. 

$16. 

91. 

52°  30'  W. 

67. 

3900  mill. 

81. 

282i|  sq.  yd. 

92. 

3  hr.  4  min. 

68. 

24  yr.  11  mo. 

6  da.  82. 

$1.15. 

93. 

m- 

69. 

Tuesday  4.  K 

>A.M.  83. 

$8. 

94. 

ill- 

70. 

Iff  ft. 

84. 

4704  ;  9408 

95. 

$29f. 

71. 

16  ft. 

shing 

les. 

97. 

7.2  sq.  Hm. 

72. 

6f  rd. 

85. 

$30.57+. 

98. 

1,000,000. 

73. 

113.0976. 

86. 

3A. 

99. 

8.1. 

74. 

4375  sq.  ft. 

87. 

A- 

100. 

42  A.  27  sq.  rd. 

75.  40  rd.  88.    99  yd.  ;  118f  yd.  ;  10|  sq.  yd. 

101.  Volume  552.9216  cu.  in.  ;  area  276.4608  sq.  in. 

102.  56  sq.  ft.  105.    73f  yd.  108.    140  sq.  rd. 

103.  $216.00.  106.    169|sq.  yd.  109.    $38.53^. 

104.  33  yd.  107.    990  sq.  ft. 

Article  285. 

1.  12  ;  12  is  3  %  of  what  ?     Atis.  400. 

2.  20  ;  20  is  what  per  cent  of  200  ?     Ans.  10  %. 

3.  $8.       6.   $.40.  9.    200.        12.   300 sheep.    15.   10%. 

4.  150.      7.   20  sheep.    10.    200.        13.    20%.  16.    40%. 

5.  45.        8.    150.  11.   $4.         14.   25%.  17.    25%;  75%. 


Article  290. 

18. 

6. 

20. 

100.            22.    16.       24.    12  gal. 

26. 

120  mi 

19. 

50. 

21. 

2  men.       23.    $6.      25.   180  pounds. 

27. 

l^in. 

Article  292. 

1. 

.07. 

6.   .0625.                11.    1.01. 

16. 

.005. 

2. 

.06. 

7.    .125.                  12.    1.10. 

17. 

.0075. 

3. 

.02. 

8.    .1575.                13.    2.50. 

18. 

.004. 

4. 

.12. 

9.    .375.                   14.    2.00. 

19. 

.00625. 

5. 

.78. 

10.   .04625.              15.    1.275. 
Article  293. 

20. 

.009. 

1. 

i- 

3.   i 

5.    I     7.    f.       9.    H.     11.    If. 

13. 

zh 

15. 

2.    i     4.    i.     6.    i.     8.    |.     10.   2|.     12.   2/5.   14.   t^.     16.    X^. 


482  ANSWERS. 

Article  294. 


1. 

2. 
3. 

4. 

.25;  i 

.60  ;  |. 
.18;  ^. 
.01 ;  T*^. 

5.  .06|;  ^.               9.    1.08;  1^. 

6.  .0625  ;  ^J^.           10.    1.50  ;  l^. 

7.  .07125;  ^V       11.    1.25;  1^. 

8.  .66| ;  |.              12.    1.375  ;  If. 

Article  295. 

13. 
14. 
15. 
16. 

.0075  ;  ^jf. 
.004;  ^i^. 
.00125  ;  ^^. 
.007  ;  TT^,. 

17. 
18. 

.$81. 

120  sheep. 

19.  61  men.                    21. 

20.  4  oranges.                22. 

2  tons. 
$60. 

Article  296. 

1. 
2. 
3. 

$48. 
200  lb. 
67.16. 

4.  6  sheep.             7.    $900. 

5.  50  bu.                 8.    128  A. 

6.  10  bu.                 9.    420  bbl. 

Article  297. 

10. 
11. 

.4. 

$5. 

7. 
8. 

$  1000. 
450  bu. 

9.    200.                  11.    2307.69+. 
10.    $4110.              12.    578. 

Article  298. 

13. 

14. 

$20,000. 

$90. 

9. 
10. 

331%. 
16|%. 

11.  89f%.              13.    8^<%. 

12.  5%.                 14.    r/o- 

Article  300. 

15 
16 

.    75 

);25%;  10o/„. 
%. 

9. 
10. 
11. 
12. 
13. 

232. 

324. 

500  sheep. 

50,000. 

$800. 

14.  $720.                19.    2001b. 

15.  644|.                20.    640  men. 

16.  $375.                21.'  $600. 

17.  $2363^.          22.    1234. 

18.  800  sheep. 

23. 
24. 
25. 
26. 

$  8000. 
2250  bu. 
$  1468.99. 

Article  301. 

1.   $20.      2.   $20.      3.   50%.      4.  12i  %.      5.  $1200.  6.   67+ cents. 

Article  303. 

7.  25%.  14.    $62.08.  21.    40%.  27.  $20. 

8.  66|%.  15.    $583.70.  22.    243^4^0/^.  28.  $240. 

9.  $22.50.  16.    $230.  23.    $172.  29.  39|  cents. 

10.  $85,695.  17.    5|  cents.  24.    $82.32.  30.  $430. 

11.  25%.  18.    36  cents.  25.    $128.  31.  150%. 

12.  9%.  19.    $1267.50.        26.   5%.  32.  $50. 

13.  $1920.  20.    $94.15. 


ANSWERS. 


433 


1.   $20. 


2.   $< 


Article  304. 
3.   $150. 


4.   $40. 


5.   $5. 


Article  311. 

6.  $43.20.  9.  $45,000.  12.   $2875.  14. 

7.  $111.56^.        10.   $1250.  13.   \%.  15. 

8.  $19.49+.  11.   $2000. 

16.  Investment,  $  5882.35 ;  commission,  $117.65. 

17.  $223.96J.         18.   750  bbl.  19.   \W. 

21.  Commission,  $2,829;  net  proceeds,  $52,371. 

22.  70  bicycles.  25.   1200  1b.  27. 

23.  1%.  26.   $120.  28. 


7  shares. 

4%. 


20.   $1052. 

$  1900. 
500  tons. 
24.   $24,459,052+.  29.   20,633^  lb.  ;  commission,  $55.71. 

30.  17,163^  A.;  commission,  $7723.50. 

31.  45,176tV  ft.  32.  7800  bu. 

34.   8000  bu.  35. 


33.   $10,706,551+. 
$5370.30. 


Article  321. 


1. 

2. 

$40. 
$3880. 

3.  $7500. 

4.  $150. 

Article  323. 

5.  $5.40. 

6.  $540. 

1. 
2. 
3. 
4. 
5. 

$90. 
$215. 
$  122.92. 
$  1418.68. 
$15. 

6.  $46.                 11.  $312. 

7.  $75.                  12.   $94. 

8.  $6.25.               13.   $30,000. 

9.  $3000.              14.   $6000. 
10.   $10,000. 

Article  326. 

15.  i%. 

16.  $5000. 

17.  $10,400. 

18.  $24,000. 

1. 
2. 
3. 
4. 
5. 

$60. 

46f%. 
$237.60. 

$42. 
$715.28. 

6.  4%.                 10.   $494. 

7.  24%.               11.   $30.40. 

8.  10%.               12.    $420. 

9.  Cost,  $342;  13.   $23.05. 

discount,  $  108. 

14.  B's    $4.32 
better  than  A's. 

15.  33i%. 

16.  80  cents. 

17.  Cost,  $  1000 ;  marked  value,  $  1266.6i 

18.  $264,024. 

Article  333. 

1.  $18,000.  6. 

2.  Rate,  .015  ;  A's  tax,  $  270. 

3.  $31.50.             4.   .02.  7. 
5.  Rate,   .016  ;   collector's  com-  9. 

mission,  $750  ;  A's  tax,  $67.50. 


Rate,  .012,  nearly  ;  B's  tax, 

$48. 

$2000.  8.   $404.61. 

$4,369,083,271.38; 

$14,155,829.80,  nearly. 


434 

ANSWERS. 

Article  338.* 

1.    $217.50. 

3.    $799.50.                       5. 

$4266.865. 

2.    $162. 

4.    $125,800.                     6. 
Article  340. 

$687,079+. 

1.    60;  $.79681;  5.^ 

ill;  6.68f.            3.   Carriage, 

$318.50;    horse, 

2.    Son,    $23,100; 

daughter,       $136.50. 

$13,750;  wife,  $3025. 

4.    100%  gain 

;  20%  loss. 

5.    $50. 

13.    3^0/^.                         20. 

$  25,000. 

6.    3280. 

14.   |.                               23. 

.00625. 

7.    Board,  $330; 

15.    $2000.                       24. 

$1237.50. 

clothes,  $198.80. 

16.    5%;  gain,  $20.        25. 

2000  bu. 

8.    726. 

17.    $2.24.                        26. 

$20. 

9.   283^. 

18.    156,250  1b.                 28. 

$180it. 

10.   $34.56. 

19.    40  cents,  original    29. 

$1.47,1^. 

11.    50.5+%. 

cost. 

12.    68|%. 

31. 

Commission,  $50.51;  net  proceeds,  $959.74. 

32.   Amt.  sales, 

,  $4222|;  commission,  $422f. 

33.    5%. 

45.    $2131.50.                  56. 

$10. 

34.   $309.37i 

46.    $10,320.                    57. 

33i%. 

35.    $150. 

47.   21,522f  bu.                58. 

Cost,  $63.95; 

36.    1%. 

48.    912  head. 

gain,  $9.59|. 

37.    $26flost. 

49.   656  plum  trees.         59. 

45%  boys. 

38.    200  boys. 

50.    55.25  gal.  sold.          60. 

14f%. 

39.   200  sheep. 

51.    595  girls.                   61. 

198  soldiers. 

40.    $6480. 

52.    9^^:%.                         62. 

100  trees. 

41.    $6814.70. 

53.   $66.25                       63. 

$500. 

42.    SOo/o. 

54.    $16,000.                    64. 

$40. 

43.    400. 

55.    .25,  nearly.               65. 

$8100. 

44.   \%. 

66.    Value  of  vessel,  $200,000  ;  A's  share,  $140,000 ;  : 

B's  share,  $24,000 

C's  share,  $9000; 

D's  share,  $  27,000. 

67.    $2.50. 

69.    88|  cents.                  71. 

90  problems. 

68.    i;  12r/o. 

70.    $3645.90. 
Article  341. 

10.    46.154. 

15. 

136.37.             20.    $371,840. 

24.    $1792. 

11.   $97,387. 

16. 

427.572.           21.    $373.99. 

25.    $1206.105. 

12.    $100.20. 

17. 

45.32.              22.    $541.35f. 

26.    $250. 

13.    $164,448. 

18. 

$347,075.        23.    $1164.435. 

27.    $646.80. 

14.    67.066. 

19. 

$387.9156. 

ANSWERS.  436 


Article  342. 


5.  $12,088.  7.  1^52.48.  9.   1 2.006.  11.    090.76. 

6.  $6.15.  --.         8.    $18.11.  10.   1. 974 J.  12.   $61.80. 

13.  Int.,  $148.83^;  Aint.,  $2498.83f 

14.  Int.,  $1,907;  Amt.,  $127,657. 

15.  Int.,  $  130.355  ;  Amt.,  $  1080.985. 

16.  Int.,  $6,309;  Amt.,  $342,789. 

17.  Int.,  $131.95;  Amt.,  $870.48. 

18.  Int.,  $110,555;  Amt.,  $5110.555. 

19.  Int.,  $103,431;  Amt.,  $970,781. 

20.  Int.,  $131,769;  Amt.,  $392,269. 

21.  Int.,  $1192.55;  Amt.,  $4242.55. 

22.  Int.,  $22,676;  Amt.,  $648,246. 

23.  $87,312.         25.   $135,708^.      26.   $520,940.        27.   $364.60. 

24.  $36^. 

Article  343. 

28.  $689,703.  32.  $327.97.  35.  $388.07.  38.  $59.25. 

29.  $853,402.  33.  $3671.938.  36.  $403.65.  39.  $315.90. 

30.  $2087.22.  34.  $870.34.  37.  $489,646.  40.  $46.66|. 

31.  $1434.426. 

Article  344. 

41.  $87.36.  45.  2+ cents.  48.  $207.43.  51.  $1,929. 

42.  $77.87.  46.  $48.93.  49.  $39.66.  52.  $1,771. 

43.  $3.5.34.  47.  $29.26.  50.  $.264.  63.  $4,422. 

44.  $42.04. 

Article  345. 

54.  $414,518.  67.    $384,259.  60.   $3560.76. 

55.  $290.50.  58.    $319,287.  61.    $186.65. 

56.  $621,317.  59.    $1953.44.  62.    $365.03. 

Article  346. 

2.  $1,198.  5.    $5,912.  8.    $102,526.         11.    $106,277. 

3.  $2,717.  6.   $9.07.  9.   $8,775.  12.    $33,418. 

4.  $14.  7.    $11.13.  10.   $21.60. 

Article  347. 

2.  Q%.  5.   40/,.  8.  8%.  10.   6%. 

3.  6%.  6.    i%.  9.    8%.  11.    6%. 

4.  9%.  7.    7|%. 


436 


ANSWERS. 


Article  348. 

3. 

3  yr.  6  mo. 

7.    4  yr.  7  mo,  6  da. 

10. 

2  yr.  16+  da. 

4. 

2  yr.  6  mo. 

8.    7  mo.  18+  da. 

11. 

3  mo. 

5. 

6  mo. 

9.    2  yr.  4  mo.  14  da. 

12. 

16f  yr. 

6. 

5  yr.  8  mo.  18  da+. 

Article  349. 

2. 

$250. 

5. 

$530.                 7.    $1730. 

9.    $980. 

3. 

$  10,000. 

6. 

$4625.                8.    $12,580. 

10.    $387.50. 

4. 

$  1000. 

Article  361. 

2. 

$245,389. 

5. 

$54,376.            7.    $478,116. 

9.    $160,177. 

3. 

$512.86. 

6. 

$406,944.           8.    $616,677. 

10.    $140,065. 

4. 

$  1380.615. 

Article  362. 


2.   $464,081. 


3.    $143,547. 


Article  363. 


2. 

$130,828.           4. 

$61,523.            6.    $56,311.            8.    $82.55. 

3. 

$761,578.          5. 

$137,924.           7.    $179,397. 
Article  364. 

8. 

6%  method.     21. 

$6635.437.       24.    $57.04.            27.    $255,223. 

15. 

$955,639.         22. 

$117.84.          25.    $51,898.          28.    $2767.30. 

20. 

$155,034.        23. 

$161,485.         26.    63+ da.            29.   6%. 

30. 

2  yr.  7  mo.  14+  da. 

38.   Dec.  1,  1895.        48.    8%. 

31. 

$ 1600f . 

39.    $436.                     49.    $500. 

32. 

$  124.999+. 

41.    $751.93.                50.  9  mo. 

33. 

$32,646. 

42.    $1258.21.              51.   6%. 

34. 

$92,001;  $715,741.      43.    $202.                     52.    $500. 

35. 

$27,193. 

44.    $35.34.                  53.    $19.60. 

36. 

$16,946. 

46.   9%.                        54.  Every  6  months. 

37. 

$  102.894. 

47.   10  mo. 
Article  368. 

8. 

Pr.  worth  $380.9523  ;     12.    $486.56+ cash.      17.    More  profitable 

$  19.0477. 

13.    $728.1553.        to  purchase  on  time. 

9. 

$2542.37+ each. 

14.    $566,037.              18.    Gain  $112.62. 

10. 

$409.0909. 

16.    Cash.                     19.   $374,987. 

11. 

$240,876. 

ANSWERS. 


487 


Article  372. 
6.   $648.13.  9.   $4918.89.        13.   $2475.  16.   $8.00. 

6.  $2962.50.        10.   $1304.57.         14.    $132,975.         17.    $249.98. 

7.  $436.24.  11.    $2168.22.         15.    $203.24.  18.    $172.55. 

8.  $781,605.        12.   $1972.44. 

19.   Day  of  maturity,  March  31,  1896.     Proceeds,  $523.99. 


Article  373. 
$253.81.  4.    $1218.27.  6.    $477.39. 

$353.53.  5.    $809.92.  7.    $495.05. 


8.   $750. 


Article  374. 

9.    $488.34.  15.    Proceeds,  $  346.56  ;  discount, 

10.  Present  value,  $  860  ;  true  dis-  $  3.44. 

count,  $113.52.  16.    Proceeds,  $864.90. 

11.  Cash,  $6.33  better.  17.    $816.33. 

12.  $168.75;  $13331.25.  18.    $209.71. 

13.  $805.37.  19.    $658,783. 

14.  $302.11.  20.    $2600. 


Article  379. 

3. 

50  shares. 

15. 

$  13,840. 

17.    $330. 

19.  $20  loss. 

14. 

20  shares. 

16. 

130  shares. 

18.  $200. 

Article  382. 

22. 

$7447.50. 

31. 

$150. 

40.  e^/o. 

48.   83f 

23. 

$  13,425. 

32. 

$200. 

41.    6|fo/o. 

49.    75. 

24. 

loir/o- 

33. 

$400. 

42.    5t\%. 

50.   75. 

25. 

12  shares. 

34. 

$11,100. 

43.  ^m%' 

51.   $80. 

26. 

20  shares. 

35. 

$5400. 

44.    ^%. 

52.    $25,500. 

27. 

30  shares. 

36. 

$9200. 

45.    6's,   n 

53.    $42,480. 

28. 

$2000. 

37. 

$  10,000. 

better. 

54.    $37,500. 

29. 

$  1010. 

38. 

$5000. 

46.    71^^. 

55.    $60,000. 

30. 

$600. 

39. 

$  10,000. 

47.    125. 

* 

Article  383. 

1. 

33^  %  premium. 

5.  Q\n- 

10. 

66f ;  140. 

2. 

$  13,600. 

6.    $16666f.                    11. 

Increase  $  25  per 

3. 

$  10,000. 

7.    $18,500.                 annum. 

4. 

6%  stock. 

/t% 

8.   133^. 

12. 

4  bonds. 

better. 


9.   6  %  bonds  at  105. 


438 


ANSWERS. 


da 


13. 

$  87,500  ;  875  shares  ;  5  %  on  investment. 

14. 

Cost,  $32,500  ;  $  1600,  yearly  income  ;  rate  on  investment,  .0492+. 

15. 

$60. 

16.    1920  shares. 

Article  387. 

2. 

84  da.  after  sale. 

5.   4|  mo.                          8. 

July  19. 

3. 

3  mo.  20  da. 

6.   4  mo.  3  da.                  9. 

5  mo.  11  da. 

4. 

Average  term,  46 

7.    94  da.                        10. 

In  7  months. 

El.;  1 

time.  May  17, 

1895. 

Article  388. 

2. 

May  4. 

4.    May  11.                       6. 

4  mo. 

3. 

Sept.  11. 

5.    41  mo.                          7. 

Feb.  7,  1897. 

8. 

Average  date,  July  13  ;  note  dated  April  13. 

Article  395. 

7. 

8. 

12. 

h           16.  m- 

19.   i;  15. 

8. 

^■5' 

13. 

48.                    17.    1 ;  4  ;  ^. 

20.   32;  7. 

9. 

h 

14. 

/^.                      18.    H;   tf; 

21.    20 ;  2. 

10. 

3. 

15. 

1-                     M;t. 

22.   |. 

11. 

4. 

Article  398. 

34. 

20,900. 

37. 

50.                   40.   $3. 

42.   25  men. 

35. 

10. 

38. 

|.                     41.    I  da. 

43.    1. 

36. 

150. 

39. 

$100. 

Article  399. 

45. 

$80^;  $21. 

51. 

$10.45.            57.    $48. 

62.   311  mi. 

46. 

3  da. 

52. 

120  Km.          58.   60  da. 

63.    $80.30. 

47. 

33f  tons. 

53. 

45  men.            59.    225  da. 

64.    15  in. 

48. 

$  5600. 

54. 

$36.                 60.    $2. 

65.    34fyd. 

49. 

85^  ft. 

55. 

$450.           *61.    240     shoe- 

66.    $27. 

50. 

12  da. 

56. 

$  3600.         makers. 
Article  400. 

67.    100  da. 

2. 

5bu. 

4.    360  mi.                        6.    •' 

$  100. 

3. 

16  da. 

5.   45,6191  stones.           7.    ^ 

t09^V  cd. 

8. 

531  rd. 

11. 

$1.84.               14.    145fm. 

17.    3  ft. 

9. 

31^  da. 

12. 

56  bu.              15.    8hr. 

18.   50  da. 

10. 

12  men. 

13. 

$3,444.            16.    16yr.8mo. 

Article  403. 

2.  Wilson,  $  900  ;  Mead,  $  600.  4.    A,    $  312^  ;    B,     $  500  ;     C, 

3.  Jones,  $  750  ;  Smith,  $  1250.     $  387|. 


ANSWERS. 


439 


5.  A,  $6§  ;  B,  $8  ;  C,  $9i.  8.    A,  §  ;  $6400  ;  B,  f  ;  $9600. 

6.  A,  $300  ;  B,  $360  ;  C,  $240.      9.    2d  =  ^  ;  1st  =  ^  ;  /=  ^. 

7.  $1800;  $2700.  10.    X,  $6250 ;  F,  $3750;  Z,  $5000. 

11.  Each  man's  gain  is  |^^,  or  -j^,  of  his  stock.  Since  j\  represents 
the  gain  of  the  first  man,  ^f  will  represent  his  stock  and  gain.  Tbe 
question  now  becomes:  $570  (A's  stock  and  gain)  is  f|  of  what? 
570  -^  if  =  $480,  A's  stock.  His  share  of  total  stock  is  ||§,  or  y%,  and 
his  share  of  total  gain  is  y%  of  $150,  or  $90.  In  like  manner,  find  the 
second  man's  stock  and  gain. 

Ans.,  Stock  of  1st,  $480.     Gain  of  1st,  $90. 
Stock  of  2d,  $320.     Gain  of  2d,  $60. 

13.  A,  $3000  ;  B,  $2100.  16.    A,    $1928//^;   B,    $1355|§|  ; 

14.  $576;  $540;  $480.  C,  $3615|f|. 

15.  A,  $1250;  B,  $2490.  17.    A,  $1440  ;  B,  $1368. 

18.  Scott,  $604,196+;  White,  $1040.559+;  Watson,  $755,244+. 

19.  A,  $2000;'  B,  $4000;  C,  21.  A,  $1373.684+ ;  B,  $1526.315. 
$1800.  22.    A,    $23.62;    B,    $20.67;    C, 


20.    A,  $95;  B,  $133. 


$30.71. 


Article  409. 


10.  7776. 

11.  1. 

12.  .00000001. 

13.  15.625. 

14.  1.21. 


15.  .000000008. 

16.  ,V. 

17.  li 

18.  15|. 

19.  1296. 


20.  3375. 

21.  1296. 

22.  243. 
24.  648. 


25.  6460. 

26.  f 

27.  .019125. 

28.  3^. 


Article  422. 


9. 
10. 
11. 
12. 


5.  94. 

6.  126. 

7.  609. 

8.  216. 

25.  f. 

26.  2.0275+. 

27.  .5773+. 

28.  |i. 

29.  6.0380+. 

30.  53.5098+. 

31.  13.7573. 
,32.  .3872+. 


906. 
5.39. 
3.04. 
56.4. 

33. 

34. 

35. 

36. 

37. 

38. 

39. 

40. 


13.  .089. 

14.  .253+. 

15.  .075. 

16.  .956. 
.4242+. 
4.1231+. 
2.1679+. 
8.5146+. 
10.9087+. 
1.9157+. 
.4711+. 
31.0322+. 


17.  5.06.  21.   41.0402+. 

18.  4.93.  22.    19742.4737+. 

19.  .782+.  23.    f. 

20.  18.1594+.  24.   |. 
41.    3.6429+.  49.    ^. 


42.  .0977+. 

43.  .0585+. 


50.  J. 

51.  .749. 


44.    58.094+ rd.      52.    6.480+. 


45.  84  rd. 

46.  .143+. 

47.  .03651+. 

48.  i. 


53.  .960+. 

54.  43.829+. 

55.  61. 


440 


ANSWERS. 


Article  426. 

3. 

32  ft. 

6. 

45  ft.                 9.    20.489+ m.      12.    7.211+ in. 

4. 

50  ft. 

7. 

339.4112  rd.    10.    117.153+ mi.    13.    61.224+ ft. 

5. 

144.5683  ft. 

8. 

42.5205  ft.       11.    20  ft. 
Article  427. 

6. 

16  to  64. 

7.   4  to  8.      8.    21  ft.       9.  26.12+  rd.      10.   320  sq.  ft 

Article  435. 

14. 

35.          16. 

16£ 

;.           18.    2.05.          20.    76.67+.           22.   3.203+. 

15. 

96.          17. 

427 

19.   2.59.          21.    12.34.             23.    216. 

24. 

.70+;. 76+; 

26. 

31.739+.          29.    1.440+.            32.    71.3  in. 

25+; 

3.39^  ;  J. 

27. 

.398+.              30.    1.912+.            33.    54  sq.  yd. 

25. 

2.428+. 

28. 

.114+.              31.    45  in.               3^    .579+. 

\ 
Article  436. 

2. 

612  cu.  in. 

4.    181.9+ lb.                      6.    13||)?. 

3. 

9.9+  in.  tliick. 

5.    7.15+ ft. 

Article  438. 

51. 

800  trees. 

53. 

$151,109.        55.    $245,898.       57.    $.20  loss. 

52. 

$6275. 

54. 

$922^.            56.    $480.               58.    18f%. 

59. 

$.51  A. 

67.   52  yd.                             76.    11. 

60. 

28f%. 

68.   46^  32' ;  133°  28'.          77.    i- 

61. 

.0911. 

69.   812igal.                        78.   /^V 

62. 

35x^^%; 

70.   2  oz.  12  pwt.                  79.    60. 

3142800.722  bbl. 

71.    1760  steps.                    80.    $307685f. 

63. 

$27.42+. 

72.    .5928+                            81.   A,   79^  mi. ;    B, 

64. 

31|  yd. 

73.    .27|.                            70^  mi. 

65. 

180  sq.  ft. 

74.   |.                                   82.    1  yr.  3  mo.  16  da. 

66. 

18.61+  ft. 

75.    $1.12.                             83.    $256,875. 

84. 

4  mo.  24  da. 

88. 

$20,000.         92.    $2336.66|.      96.   Cash. 

85. 

$200. 

89. 

$440.83^.        93.    $2060.            97.   $1152.52. 

86. 

$200,518. 

90. 

$337,534.        94.    $450.              98.    $2470.83^. 

87. 

$725.07. 

91. 

$83.82.            95.    $1.8349.         99.    $.215. 

100. 

A,  $300  ;  B,  $297.                        104.    60  da.,  Mar.  4,  1890. 

101. 

$48,346. 

105.    $570. 

102. 

June  8,  '95. 

106.   12  bonds. 

103. 

$75,037+. 

ANSWERS.  441 

107.  Simple  int.,  $  1300  ;  4%  bonds,  $  1214.44  ;  5%  bonds,  $  1200. 

108.  $355,020.57+.  114.  |  ;  |^.  120.  if. 

109.  37|%.  115.  8  horses.  121.  $316.64+. 

110.  6|%.  116.  l  122.  6. 

111.  6%,  at  125.  117.  IjW/oft.  123.  $12. 

112.  Lost  $62.50.  118.  8f  oz.  124.  $  96,  $  60,  $  144. 

113.  180  1b.  12  oz.  119.  12  hr.  125.  87 1^  on  the  dollar;  $3360. 

126.  $  2105^f  ;  ^  1578f  f  ;  $  12ii3^^j  ;  $  1052f  f . 

127.  $240;  -$120;  $80. 

128.  Scranton,  $  1555  ;  Morris,  $930  ;  Jackson,  $620. 

129.  $12;  $28;  $20.  130.    5.656+ ft. 

131.  Shorter  sides,  17.204+  ft. ;  longer  sides,  18.867+  ft. ;  room,  23.494  ft. 

132.  21  ft.  137.    6.3+  ft.  142.  240  rd. ;  $  588. 

133.  24.495+ ft.  138.   210  sq.  ft.  143.  24,200  revolutions. 

134.  $3.8816+.  139.    34  ft.  144.  2%. 

135.  34.7.  140.    $11,120.89|.  145.  $213,334. 

136.  6463.8-»  141.    69.641ft.  146.  $823.20. 
147.  2567%.  148.    1%.              149.  $94.15.  150.   $80.75. 

Article  440. 
6.   99  bu.  7.    775^.  8.   376.  9.   240 L. CM. 

Article  441. 

3.  $8.95.  5.    575  bbl.  7.   13  yr.  10.   37  mi. 

4.  $5340.  6.    11,  7,  5,  3.        8.    1560  bu. 

Article  442. 

1.  Twenty-three  million  four  hundred  fifty-six  thousand  seven  hundred 
eighty-nine. 

2.  G.  C.  D.,  25  ;  L.  C.  M.,  1500.  3.    $43.35. 

4.  30,060,290.  6.   2,  2,  3,  11,  11.  8.    72. 

5.  60  qt.  7.   $19.00.  9.    60001b. 

Article  443. 

1.  456  mi.  4.    $1750.  7.  f. 

2.  204,060,402.  5.   5182.  8.  19  marbles. 

3.  987.  6.    $1060  gain.  9.  150  eggs. 

Article  444. 

1.  45  J?.  4.    ■5^.  6.    21  f.  9.    ^ij^. 

2.  8yr.  5.    $3^.  7.    $9.00.  10.    i^^^,or5^. 


442 


ANSWERS. 


Article  445. 

1.  $84,000.  3.    Increased!.  5.  f. 

2.  2tV  da.                      4.    27,853.  6.  7|§  cords. 
7.    2,  3,  7,  3,  2,  11,  5.                  8.    1.  9.  78. 


1.  MDCCCXCIV. 

3.  9  boxes. 

4.  160  sheep. 
9.  hhhh 


1.  $495.63. 

2.  450  sheep. 


Article  446. 

2.    1,  2,  3,  5,  7,  11,  13,  17. 

5.  mi  7.    $34^^. 

6.  4ibbl.  8.    8|. 


3.  2||. 

4.  |. 


Article  447. 

5.  63  cows. 

6.  1511. 


10.    |. 


9.    $6611.11|. 
10.    $3.04. 


2.  Hi 

3.  h 


Article  448. 

4.  B,  H,  M-  6.    17^^.  8.   24^5^. 

5.  $f  7.    1.  10.   $15,975. 


2.  271^  yd. 

3.  2. 

4.  if  lb. 


5.  9  persons. 

6.  20. 


Article  449. 

7.    77^ 


8.   3fda. 


1.  $4608.  4.   f. 

2.  Increased  ^83.    5.   7y\. 

3.  5V  6.   23ibu. 


Article  450. 

8.  32|, 

9.  y-i. 


Article  451. 


1.  61f  cents.  3.   42^f§. 

2.  32^  mi.  4.    20f  |  da. 

7.    One  thousand  nine  hundred  fifty 
seventy-three  million  seven. 
9.    $4345.  10.    $187. 


Article  452. 
3.   43HA. 


4.    2 -2.  2.  2.  3.  3.  7. 


9.    36  ft. 
10.    $4900. 


10.  m^M^ 

188      150      160 
ttt^  ttS^  Jt^' 


5.  mi 

6.    29,  707,  578. 
ninety  ;  four  thousand  forty  ; 


1.  157|fyd. 

2.  6.225. 

7.  $27,168+. 

8.  Sixty-eight  and  six  hundred  forty-two  ten-thousandths. 

9.  $1.95.  10.    $22.75. 


5.  1000. 

6.  $1.1856. 


ANSWERS.  443 


Article  453. 


1.  $1200.  3.    hf. 

2.  999999.999999.  5.    7,095,000  ;  63.015  ;  700.07.' 

6.  Six  hundred  forty-two  and  sixteen  ten-thousandths ;  one  hundred 
and  one  hundredth. 

7.  600700.19+.  8.    8295.9056.  9.    $60,552.  10.    $8.06. 


Article  454. 

1. 

2. 
3. 

\\\.                4.    $.1875.              6.    192  1b. 
%.                   5.   216  sheep.         7.   23^3  da. 

Article  455. 

8.    $250. 
10.    $161.30f. 

1. 
2. 
3. 
4. 
5. 

^1  is  3^6^  greater.                          6.    .013;  400.05; 
250^.                                            7.    ^V.. 
60  ft.;  19  lots.                              8.    $3731. 
$6^;  12|T.                                 9.    $4.58.' 
.75;  .8;  .625;  .85;  .0555+.       10.    $4.26. 

Article  456. 

.000515 ;  400.063? 

1. 
2. 
3. 

3^.                  4.    .4375.                 7.    .05^^^. 
1,000,000.      5.    X^-^-^^.                8.    .8f|. 
3.43.               6.    1.0419479325. 

9.0. 
10.    .6559. 

Article  457. 

1.  80.65. 

2.  Ten  thousand  twenty  and  forty-two  thousand  twenty-four  hundred- 
millionths  ;  seven  hundred  two  millionths  ;  eighteen  hundred-millionths  ; 
thirty  thousand  and  thirty  hundred-thousandths ;  ten  thousand  twenty 
hundred-millionths. 

3.  2da.  2hr.      5.    220.89906.  7.    $408.54.  9.    .22|. 

4.  $2.85.  6.    999.999.  8.    ^^.  10.    1000  1b. 


Article  458. 

1. 

$38.                4.    118-/g.                 7.    2V 

9. 

$2912. 

2. 

240.464501.    5.    $5376.70.           8.    .904. 

10. 

$  24.64. 

3. 

2.26;2^2j<V.     6.    $13.40. 

Article  459. 

2. 

88  T.                                          5.    80  sq.  yd. 

7. 

Q>^  sq.  yd. 

3. 

2  lb.  5  oz.  18  pwt.  3  gr.          6.    54  sq.  ft. 

9. 

\%  ;  h  ;  iV 

444 


ANSWERS. 


Article  460. 

1.    I  or  1|.       2.    15  da.       3.   4  fur.  8  rd.  1  yd.  .7225  ft.       4.  4^  sq.  yd. 

5.  6  A.  118  sq,  rd.  24  sq.  yd.  2  sq.  ft.  22f  sq.  in. 

6.  $16.91|;  $9.  7.    10  A.  8.   216. 


1.  .754+. 

2.  9  T.  16  cwt.  61  lb. 
12|  oz. 

3-    -sT^W^Tiru* 


Article  461. 

4.  216^  yd. 

5.  14  oz.  2pwt.  23fgr. 

6.  1513.69875. 


Article  462. 


1.  $1.22.  4.    3pk.  4qt.  7.    $67.46|. 

2.  $287^1^.  5.    172fbu.  8.    $46.40. 

3.  $16.  6.    $261. 

Article  463. 


39|^. 

280  rd.  2  yd.  4  in.- 

248  boards. 


1.  518  pickets. 

2.  50  rd.  X  25  rd. 

3.  74  to  the   square 
mile ;  8|f  A. 


6.  m- 

7.  4281^V 

8.  18  da. 


Article  464. 

3°  3';   12  min. 
sec.  before  12. 
.$31.16|. 
$9. 


12 


7.  $5237.50. 

8.  $161.77. 
10.    16,995  far. 


9.    20°  East. 
10.    19fbbl. 


9. 


10. 


9  birthdays  ;  36 
yr.  16  da. 
.000000144. 


7.  $32.40. 

8.  480  boards. 

9.  16^. 
10.  34|qt. 


1.  $374. 

2.  $7200. 

3.  6  times  ;  1  qt. 

4.  $105.60. 

1.  32  blocks. 

2.  42  families. 


Article  465. 

$84. 
$6,451. 
1161  liters. 

Article  466. 


20  pairs. 
12  mo. 


8.  64,800  bricks ;  $476.28. 

9.  645,126  cu.  in. 
10.    114.5+ A. 


24 1  gal. 


m- 


9. 
10. 


1.  1368    cu.    ft.; 
$  48.09f . 

2.  15.915+  ft. 

3.  $.05. 


Article  467. 

$4.45^^. 
$  12.0645. 
$  5.325. 
16  da. 


9.    5  mi.  143  rd.  2  yd. 
7^  in. 
10.    128  mi.  281   rd. 
1  yd.  2  ft.  2  in. 


ANSWERS. 


445 


2. 


1  fur.  85  rd.  3  yd. 

2  in. 

5  bu.  1  pk.  4  qt. 
1  pt. 


Article  468. 

3.  4  yr.  7  mo.  18  da. 

4.  $46.09^t;  $68.68^. 

5.  8^1^  acres. 

6.  $7.04. 


7.  $13.60. 

8.  35,000  gr. 

9.  .64  ton. 
10.  576||. 


Article  469. 


1.  28%;  610/,;  307o/„;  |o/^. 

2.  .005;  .0625;  .08;  1.25. 

3.  \^%. 

4.  $.10. 

5.  $432. 


6.  18f<%. 

7.  786  bbl. 

8.  $  18.42  gain. 

9.  38^%. 

10.  Rate,  .0135;  A's  tax  $51. 975. 


1.  28fhr. 

2.  If 

4.    .50  ;  .25  ;  .20 ;  .10. 


Article  470. 

5.  201.84. 

6.  i;f;  i;i; 

7.  324  bbl. 


8.  200  sheep. 

9.  $5300. 
10.    $206.85. 


1.  Neither. 

2.  20%. 

3.  33^%. 
5.  $67.60. 


1.  $3,705. 

2.  $5600. 

3.  $8000. 


Article  471. 

6.  $6000. 

7.  Rate,  .002  ;  A'j 
tax  $  10. 

Article  472. 

4.  $637.50. 

5.  150  1b. 

6.  $1301.86575. 


8.  $119.90. 

9.  405. 
10.    $8240. 


9.    $20.81^,. 
10.   2.61+  ft. 


Article  473. 


1.  $  .03 ;  4  bu.  ;  120  eggs ;  16  cwt. 

2.  M- 

3.  Not  changed. 

5.  13tV^%;  14r/o;200o/„;  10/^. 


6.  .0025;  .2;  .0025;  .20;  .155. 

7.  93H%. 

8.  $7200. 

9.  lOo/o. 
10.  $612.85. 


Article  474. 

1.  Rate,  .0042.  3.    $4000.  ,  6.   4ff. 

2.  Gain,  2f%.  4.    $32iV  7.    $30.80. 

8.    1\  gal.  =  1  cu.  ft.;  4810 gal.  9.    $42.1875.  10.    Gram. 


446  ANSWERS. 

Article  475. 
1.    11820.  2.   $2740.50. 

6.  Two  million  three  hundred  thousand  four  hundred  six  and  nine 
hundred  sixty  millionths. 

7.  Denominate;  8.    $60,000.  9.    $1920,  gain.  10.  33^%. 
Mixed. 

Article  476. 

1.  .02;  06J;  .20;  .121  3.    I;  |;  l;  1;  If. 

2.  20%;  75%;  121%;  i40o/„.  4.    ^h  I  ■5h  I  jh  ;  sh  •:  t^^o. 

5.  10  tons.  7.    80%.  9.    1311. 

6.  $  778.05.  8.    614  days.  10.    100i|  cu.  ft. 

Article  477. 

2.  Eighty-three  and  four  million  nine  hundred  thirty-seven  thousand 
seven  and  ^  ten-raillionths  ;  one  million  one  thousand  one  and  one 
hundredth ;  ninety  thousand  nineteen  and  four  hundredths. 

3.  240.  8.    27.8;  .1875;  5.2125. 

5.    115  lb.  11  oz.  5  pwt.  9.    $2  per  bushel ;  100%,  profit. 


6. 

If 

10.    3  0; 
Article  478. 

'o- 

1. 
2. 
9. 

$  1436.40. 
3750  lb. 
15%;  61%; 

3. 

4. 

50% 

20%.                  5.    2iffl|, 
18,760  ft.          6.    22|% 
;  226  o/o. 

Article  479. 

;  50 
10. 

7.    $4. 
1%;  6210/,.          8.    $24. 

ill;  .V.;l;if. 

1. 

2. 
3. 
4. 

$  12,000. 

$  .63^. 

io|%. 

188  ;  376  ;  376. 

5.  $371.25. 

6.  Mj%Y/o- 

7.  32H%. 

Article  480. 

8.  $10,260. 

9.  $909f 
10.    21ffo/,. 

1. 
2. 
3. 

$946.75. 

.$6,037. 

$4.00. 

6.  $3,161. 

7.  117. 

Article  481. 

8.  165.68;  $1365.60. 

9.  $36.166f. 

1. 

3  yr.  10  mo 

.20  da.           2.    8%.           5.    i 

$6523.08.          9.    $12,000. 

Article  482. 

1. 
2. 
3. 

$544.05. 
1  yr.  6  mo. 

m%- 

4.  611  cu.  yd. 

5.  100%;   1. 

6.  .06  ;  1.06. 

7.  216  farms. 

8.  705. 

9.  $598,978. 

ANSWERS. 


447 


1.  $17.07. 

2.  $697.00. 


Article  483. 


10  mo. 


5.    $350,957. 
8.    $844,216. 


9.    $185. 
10.    $104,413. 


1.  §845.021. 

2.  $669.11/^. 
4.   $12,753. 


Article  484. 

5.  $1423.327+.  8.    25  yr. 

6.  6%.  9.    $42,169+. 

7.  1  yr.  6  mo.  18  da.  10.   $367^. 


4.  $622,425. 

5.  $512. 


Article  485. 


6.  410/0. 

7.  3  yr.  5  mo. 


8.  $606.08. 

9.  $128,865. 


10.    A,  $1704;  B, 
$1597|;  C,  $958^. 


1.  $25. 

2.  2d;  $308. 

3.  $77.29. 

4.  $1812  gain. 


Article  486. 

5.  $3270. 

6.  $542,955. 

7.  Int.,  $2145.282; 
amt.,  $9376.57. 


8.  $742.50. 

9.  Matures    July    2, 
1896;  proceeds,  $595.70. 

10.    $8685. 


Article  487. 

1.    31|.  2.    Both  cost  $4848^1 ;  lost,  $48ff. 

3.  $4240.10.  5.    $2625.  7.    30%.  9.    9200. 

4.  $21,600.  6.    $1010.  8.  $8000.  10.    $10,000. 


1.  16153.84. 

3.  $225. 

4.  $15817.75. 

5.  6%  bonds. 


Article  488. 

6.  $3030.32. 

7.  Average  term,  62 
da. ;  equated  time,  Mar.  4. 

Article  489. 


8.  Nov.  30. 

9.  250  shares. 

10.    5  %  per  annum, 


6.  Present  worth,  $285.11 ;  discount,  $14.29. 

7.  Bank  discount,  $10.95;  proceeds,  $619.05. 

8.  $3000.  9.    \.  10.    150. 


Article  490 

6.    3,  missing  terra. 

8.    U}  da. 

7.    6|da. 

9.    2  hr.  45  min. 

10.    $367^. 


448  ANSWERS. 

Article  491. 
1.   24  men.  2.    A,  $540;  B,  $300. 

3.  64  cents  on  a  dollar;   A,  $224;    B,  $435.52;    C,  $41.60;   D,  $320: 
E,  $627.52. 

4.  I;  f ;  li  7.    A,  $14.40;  B,  $25.60.  9.    if. 

5.  4  da.  8.    1  yr.  6  mo.  10.   f  |. 

6.  21  da. 

Article  492. 

1.  A,  $240;  B,  $280;  C,  $300. 

2.  A,  $18,500;  B,  $24,800;  C,  $20,700. 

3.  B,  $6000;  C,  $7500.  4.    $90;  $120;  $240. 

5.  A,  $600;  B,  $5331;  c,  $266|. 

6.  Foster,  $200;  Stull,  $180;  Furlong,  $120. 

7.  $10,000;  $3500;  $2000;  $2500. 

8.  A,  $4000;  B,  $1600;  C,  $2400. 

9.  A,  $3500;  B,  $5000;  C,  $7500. 

10.    Smith,  $2500;  Brown,  $2700;  Jones,  $4200. 


Article  493. 

2. 
3. 
4. 

1. 
3. 
4. 

V^  ;  5.5225.         5. 
1,860,867.             6. 

169.7+  rd.  X  56.57- 
109.54+  ft.            5. 
15.362+  ft.            6. 

320  rd.            7.    201  in. 
$576.              8.    10,584  sq. 

Article  494. 

-  rd.                               2.    4^  ] 
3|  A.               7.    132  ft. 
56.56+ rd.       8.    21.63+ ft. 

in. 
mi. 

9.   66.5+  in. 
10.    12.7- ft. 

9.    15.3.69  mi+. 
10.    34||  cents. 

Article  495. 

1. 
2. 
3. 
4. 

1529;  591. 

29.90. 

$11,440. 

$75,997. 

5. 

6. 

7. 

Other  number,  53| ; 
912.4|. 

$25,568^23-. 
600. 

Article  496. 

8.  $240. 

9.  1611. 
10.   5. 

1. 
2. 
3. 

1,425,600  ties. 
2  hr.  253^  min. 
13li|. 

4. 
5. 

$1509.                  6.    8^. 
1650  times.           7.    90  da. 

Article  497. 

9.    $4H. 
10.   3rVda. 

1. 
3. 
4. 

$.84;  6  marbles. 
30.                         5. 
448.                        6. 

2.    875  examined ; 
$80,000.           7.    48. 
$12,000.           8.    2  da. 

625  passed. 

9.   36  fish. 
10.    20  da. 

ANSWERS.  449 

Article  498. 

1.  $73.50.  5.  ^  cords.  8.  $18.48. 

2.  1141  yd.  6.  ^-^  hhd.  9.  50  bd.  ft. 

3.  f  7.  280.9078  sq.  in.  10.  46  min.  28^  sec.  past  9,  a.m. 

4.  $52.92. 

Article  499. 


1. 

2. 
3. 

7. 
8. 

1  min.  48  sec.  past  6,  a.m. 

5  ft. 

106  sq.  rd.  25  sq.  yd.  4  sq.  ft.  126  sq.  in. 

1  hr.  14  min. 

51  ft.  5^  in.  long,  16  ft.  wide. 

4. 
5. 
6. 
9. 
10. 

$52.44. 
.803+. 
1704  sq.  in. 
47  If  bbl. 
$2.07|. 

Article  500. 

1. 
2. 
3. 

60%. 

4. 
5. 
6. 

$900,088.               7.    2000  bbl. 
1787;  1413.           8.    14%. 

$757.76. 

9.    .46^^o/^. 
10.   2%. 

Article  501. 

2. 
3. 
4. 

$  1458.465. 
80  ^. 
$612. 

1.    Rate,  .02  ;  A's  tax,  $  125. 

5.  $36. 

6.  $3500. 

7.  Direct,  $2193  better. 

8. 

9. 

10. 

$4.57|. 
$  2000. 
$  7920. 

Article  502. 

1. 

Gained  $293| 

2.    28^. 

3. 

1|  months. 

4.    Discount,  $  9.80  ;  Proceeds,  $1213.20. 

5.  A,  $18^;  B,  $25;  C,  16f.  7.   f|. 

6.  llOjVft.  8.    3|. 

9.   A,  $2500;   B,  $1500;   C,  $1000.      A's  gain,  $1000;   B's  gain, 
$600;   C's  gain,  $400. 

Article  503. 
1.   11^  bbl.  2.    135.66+  ft. 

3.  Stevens,  $86f|  ;  Jones,  $lllff  ;  Payne,  $37m. 

4.  165  shares. 

5.  A,  $  149.33  ;  B,  $224  ;  C,  $  186.66f. 

6.  100  rd.  9.    J. 

7.  4  %  stock,  yf^  %  higher.  10.    $240. 

8.  Wife,  \ ;  daughter,  I ;  sou,  \  each. 


450  ANSWERS. 

Article  504. 

1.  $1.60.         3.    $260.  5.    144.         7.    24;  \.  9.    $2000. 

2.  80.  4.    23  men.  6.    10.  8.   3^%.  10.    1|%. 

Article  505. 

3.  6  P.M. ;  8  P.M. ;  2.24  p.m.  8.    3f 

4.  fhr.  9.    $93.75. 

6.  A,  $40;  B,  $50.  10.    14.14+ w. 

7.  8|f|  ft.  wide. 

Article  516.     Mensuration. 
1.    432  sq.  rd.  2.    120  sq.  ft.  3.    150  sq.  ft. 

Article  518. 

4.    175  sq.ft.  5.    21i  bd.  ft. 

Article  520. 

6.    72  sq.  ft.  7.    29^25  A. 

Article  522. 
8.    70  sq.  in.  9.    480  sq.  in.  10.   150  sq.  ft. 

Article  525. 

11.  160  cu.  in.  13.    144  sq.  ft. 

12.  192  sq.  in.  14.    90.478+  sq.  in. ;  100.5312  cu.  in. 

Article  527. 

15.    380  cu.  ft.  16.    63.879+  cu.  ft. 


Article  530. 

17. 

804.504+  sq.  in. 

22.    7238.2464  cu.  ft. 

18. 

113.0976  sq.  yd. 

23.    33.5104  cu.  in. 

19. 

314.16  cu.  cm. 

24.    113.0976  cu.  dm. 

20. 

8181.25  cu.  ft. 

25.    Cube,  64  cu.  ft. ;  sphere,  33.5104  cu.  ft. 

21. 

33.5104  cu.  ft. 

26.    Cube,  96  sq.  ft. ;   sphere,  50.2656  sq.  ft. 
Article  531. 

1. 

11.78  cu.  yd. 

2.    307.8768  sq.  ft.                3.    8  sq.  ft. 

4. 

Area  of  square,  5625  sq. 

ft. ;  area  of  circle,  7162+  sq.  ft. 

5. 

28.2744  sq.  ft. 

9.    84  sq.m.                         12.    $26.39. 

6. 

$3500. 

10.    48  cu.  ft.                        13.    1484.40  gal. 

7. 

596,904,000  miles. 

11.    137.922  cu.  ft.                14.    62.832  Kl. 

8. 

25.3+  in. 

ANSWERS. 


451 


3.  $5075. 

4.  $494.61. 

5.  $3045. 

6.  $878.38. 

3.  $1873.915+. 

5.  $154.71. 


Article  548.     Appendix. 

7.    $731.25.  11.    .$504.75. 

9.    $984.50.  13.    $1000. 

10.    $4997.50.  14.    $2261.306. 

Article  553. 
6.    $246. 
8.    12875  francs. 


16.  $2552.322. 

17.  $3519.354. 

18.  $2036.483. 


9.    £1000. 
10.  4000  marks. 


t(^i 


( 


THIS  BOOK  IS  DUE  ON  THF.  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL   BE   ASSESSED    FOR    FAILURE  TO    RETURN 
THIS    BOOK   ON    THE   DATE   DUE.    THE   PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY    AND    TO     $1.00    ON     THE    SEVENTH     DAY 
OVERDUE. 

Map   i?o  «/x 

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0 

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0^  Q\  ^^^ 

,  1     V      "I  A    '  f        IV 

'■'    '-^     X'O  I       f 

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LD  21-100m-7,'39(402 

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